# Coupled Delft3D-Object Model to Predict Mobility of Munition on Sandy Seafloor

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{opb}); (b) a sediment scour model with the sediment Shields parameter (θ

_{sed}); (c) an object’s physical parameters such as diameter (D), relative density versus water density (S

_{o}), mass (M), and rolling moment about its symmetric axis (I

_{o}); (d) environmental variables such as near seabed ocean currents, bottom wave orbital velocity (U

_{br}), water depth (h), wave peak period (T

_{P}), significant wave height (H

_{S}), and sediment characteristics; (e) the model output such as the burial percentage p

_{B}, and the object’s displacement.

## 2. Study Area

## 3. TREX13

#### 3.1. Surrogate Munitions

#### 3.2. Field Experiment

#### 3.3. Data

_{50}) is around 0.23 mm, and the sediment density (ρ

_{s}) is about 2.69 × 10

^{3}kg m

^{−3}. These two parameters are most significant to influencing sediment mobility and are needed for the object scour burial model (see Section 6).

## 4. Delft3D

#### 4.1. Model Description

_{P}), significant wave height (H

_{S}), and bottom wave orbital velocity (U

_{br}), with a temporal resolution of 1 h for the object model, the coupling time between the flow and wave modules was also set to 1 h. The morphology module works in an integrated way with the wave and flow modules in a cycle. This system is a process-based model that considers the impact of waves, currents, and sediment transport on morphological changes. In this study, the sediment module was not activated.

#### 4.2. Model Grids and Time Steps

#### 4.3. Wind and Tidal Forcing

#### 4.4. Initial and Boundary Conditions

^{3}. The initial condition for the sand sediment was set as a uniform zero concentration, and the initial bed of sediment was set to 5 m. Wave boundary conditions were set based on the measurements from the deep quadpod location using the significant wave height, wave period, wave directions, and directional spreading. These parameters were applied uniformly on the three open boundaries. A spin-up interval of 720 min was established to prevent any influence from a possible initial hydrodynamic instability on the bottom change calculation, which starts only after the spin-up interval. The sediment type was set as sand with a sediment-specific density of 2650 kg/m

^{3}.

^{2}/s

^{3}and the wave height to water depth ratio was 0.7, for the wave module. The Chézy bottom roughness was 65 m

^{1/2}/s, horizontal eddy viscosity was 0.5 m

^{2}/s, and horizontal diffusivity was 10 m

^{2}/s, for the flow module.

#### 4.5. Model Output

**U**

_{c}=

**i**v

_{e}+

**j**v

_{n}, with (

**i**,

**j**) the unit vectors in longitudinal and latitudinal directions, and U

_{c}= (v

_{e}

^{2}+ v

_{n}

^{2})

^{1/2}as the current speed. The output from the wave module includes the wave peak-period (T

_{P}), significant wave height (H

_{S}), wave direction, and bottom wave orbital velocity (U

_{br}). The bottom water velocity vector of combined current and waves is represented by

**V**

_{w}with |

**V**

_{w}| = U

_{c}+ U

_{br}and the orientation ψ = tan

^{−1}(v

_{n}/v

_{e}). Figure 7 shows the time series of the environmental parameters [v

_{e}, v

_{n}, h, T

_{P}, H

_{S}, U

_{br}] predicted by the Delft3D (red curve) and observed by the AWAC (black curve). The AWAC only provides the observed data for [v

_{e}, v

_{n}, h, T

_{P}, H

_{S}], but not the bottom orbital velocity U

_{br}, which was calculated using a well-established linear wave model with a Matlab function [21] using the observed water depth (h), significant wave height (H

_{S}), and peak period (T

_{p}) (see Appendix D in [21]).

## 5. Object Mobility Model

**V**

_{w}) be in the direction towards the cylinder with an angle, ϕ, perpendicular to the main axis of the cylinder and be decomposed into

**V**

_{w}= (U, V), with U being the perpendicular component and V the parallel component (Figure 8) to the main axis of the cylinder. As the object rolls with angular velocity $\omega $ on the seabed with the object’s burial, let the axis of rotation inside the sediment be at depth b (b < B) (see Appendix A).

_{o}is the rolling moment of the munition about the symmetric axis of the munition (see Figure A2); T

_{F}is the forward torque caused by the drag force (F

_{d}) and lift force (F

_{l}) (see Appendix C); p

_{B}= B/D, is the percentage burial, and θ

_{opb}is the object mobility parameter for percentage burial (see Appendix B); (ρ

_{o}, ρ

_{w}) are the densities of the object and water; Π is the volume of the munition.

_{k}to t

_{k}+ 1 (k = 0, 1, 2,…, K−1) [23]

_{k}and β

_{k}as known constants during the integration. Substitution of (6) into (3) leads to the dimensional horizontal velocity of the rolling object, u

_{o}(t) = U${\widehat{u}}_{o}$(t), which should be used for each time interval Δt. The solution (6) depends on (α

_{k}, β

_{k}) which involve three types of parameters: (a) time-independent physical parameters of the object for S

_{o}, Π, L, and D; (b) time-dependent water velocity, U(t

_{k}), from the Delft3D model output; (c) time-dependent relative depth of sediment rolling axis [p

_{b}(t

_{k})] and burial percentage [p

_{B}(t

_{k})] determined using a scour burial model. Let l be the displacement of the object,

## 6. Object Scour Model

_{sed}),

_{B}(t) = B/D [15,16]. Here, f is the wave friction factor [24], ρ

_{s}the sediment grain density, and d

_{50}the medium sand grain size. Using the wave data (T

_{P}, U

_{br}) from Figure 6 e.g., and sediment parameters (ρ

_{s}= 2.69 × 103 kg/m

^{3}, d

_{50}= 0.23 × 10

^{−3}m) from TREX13 [9,13], the sediment Shields parameters (θ

_{sed}) were calculated from 21 April to 7 May 2013. The results were less than 0.1 every time, except when atmospheric cold fronts passed by on 5–6 May 2013. The maximum value of θ

_{sed}reached 0.33 (Figure 9).

_{B,eq}for motionless cylinders induced by scouring tends to increase as θ

_{sed}increases. An empirical formula has been established,

_{1}, a

_{2}, a

_{3}) determined experimentally for cylinders subject to steady currents: a

_{1}= 11, a

_{2}= 0.5, a

_{3}= 1.73 [25], a

_{1}= 0.7, a

_{2}= a

_{3}= 0 [26], a

_{1}= 2, a

_{2}= 0.8, and a

_{3}= 0 [27]. For cylinders under waves (depending on wave period): a

_{1}= 1.6, a

_{2}= 0.85, and a

_{3}= 0 for T

_{p}longer than 4 s [28]. For motionless cylinders, before scour burial reaches equilibrium, the percentage burial follows an exponential relationship [25],

_{br}), sediment density(ρ

_{s}), medium grain size (d

_{50}), and in turn the sediment Shields parameter (θ

_{sed}), the equilibrium object percentage burial (p

_{B,eq}) is calculated using (9) with coefficients a

_{1}= 1.6, a

_{2}= 0.85, and a

_{3}= 0. The sediment supporting depth b (or p

_{b}) is calculated from burial depth B (or p

_{B}) using (10), i.e.,

_{B}) computed from (10) represents the depth that an object on the surface would bury to at that moment, but an object deployed at the beginning of the time sequence would always remain buried at the deepest burial it has reached so far. The burial depth of the base of the object below the ambient seabed is equivalent to the greatest depth that the scour pit has reached up to that point in time [29]. In other words, scouring only acts to bury an object deeper. It can never be unburied (re-exposure), as the time series is suggested by (10). Similar to [29], a simple parameterization was proposed [9] to represent the re-exposure process starting from k (= 1, 2, …): (a) doing nothing if ${p}_{B}({t}_{k+1})\ge {p}_{B}({t}_{k})$, (b) computing the weighted average

## 7. Prediction of Object’s Mobility and Burial

**V**

_{w}(data represented by the red curves in panels a, b in Figure 7) and the direction perpendicular to cylinder’s main axis is determined. The velocity vector of combined current and waves (

**V**

_{w}) is then transformed into

**V**

_{w}= (U, V) with U being the perpendicular component, and V the paralleling component. The component U is used in the model. The object’s physical parameters, such as the diameter (D), volume, mass (M), and density (ρ

_{o}), are obtained from Table 1.

_{P}), significant wave height (H

_{S}), bottom wave orbital velocity (U

_{br}) (represented by the red curves in Figure 7), and sediment data (100% sand, d

_{50}= 0.23 mm, ρ

_{s}= 2.69 × 10

^{3}kg m

^{−3}) are used by the sediment scour model (i.e., Equations (8)–(13)) to get the burial percentage p

_{B}(t

_{k}), and in turn the relative rolling center depth p

_{b}(t

_{k}). With the object’s physical parameters (D, ρ

_{o}, M), the calculated p

_{b}(t

_{k}), model-predicted bottom current velocity component perpendicular to the object’s main axis U(t

_{k}), and the coefficients [α(t

_{k}), β(t

_{k})] for the object mobility model (i.e., Equation (5)), the object’s displacement at the next time step l(t

_{k}

_{+ 1}) can be predicted using (7).

_{B}(t

_{k})) shown in Figure 10, the objects’ mobility parameter for percentage burial (θ

_{opb}(t

_{k})) shown in Figure 11, and the objects’ displacement (l(t

_{k})) shown in Figure 12. The burial percentages p

_{B}for all the objects were less than 0.5, except during the storm event at 12:00 on 5 May to 00:00 on 6 May 2013 local time (Figure 10). The red color in Figure 11 shows that the objects’ rolling condition [θ

_{opb}> 1] is satisfied.

## 8. Conclusions

- (1)
- A coupled Delft3D-object model was recently developed to predict underwater cylindrical objects’ mobility and burial in a sandy bed. The roll of the object is the major dynamic of this model, with a new concept of its rolling center in the sediment. The object’s displacement caused by rolling satisfies the Riccati equation with an analytical solution. Along with the dynamical model, the empirical scour model with re-exposure parameterization is used as part of the prediction system.
- (2)
- Data collected at the shallow quadpod during TREX13 (21 April to 23 May 2013) off the coast of Panama City, Florida were used as model verification. The environmental data, such as bottom currents, water depth (h), peak period (T
_{p}), and significant wave height (H_{S}), are used to verify the Delft3D model. The objects’ positions tracked by sector scanning sonar images and maintenance divers are used to verify the object’s mobility and burial model. - (3)
- The model predicted object positions agree qualitatively well with the observed surrogates (or replicas) data. Observation shows that objects A2 and C2 were immediately mobile and transported out of the field of view, because they were last seen on 23 April 2013. The other objects were nearly immobile. The objects with large mobility are A2 (displaced 20.7 m from 12:00 21 April to 12:00 24 April 2013) and C2 (displaced 6.52 m for C2 from 12:00 21 April to 00:00 23 April 2013). A2 is a 20 mm cartridge with a mass of 0.11 kg and a density of 1429 kg m
^{−3}. C2 is an 81 mm mortar with a mass of 1.45 kg and a density of 1.199 kg m^{−3}(see Table 1). The other objects, with almost no mobility, are A5 (density of 2597 kg m^{−3}), B5 (density of 2356 kg m^{−3}), C4 (density of 3109 kg m^{−3}), C6 (density of 7194 kg m^{−3}), D3 (density of 2721 kg m^{−3}), and D6 (density of 4444 kg m^{−3}). The larger the object’s density, the smaller the object’s mobility parameter for percentage burial (see Equation (A14)). However, it is noted that the observational object data are quite crude and not sufficient to accurately verify the model prediction on an object’s mobility and burial. - (4)
- Although the coupled Delft3D-object model has the capability to predict an object’s mobility, the model has its own weakness specifically regarding cylindrical objects. It only considers the roll of the cylinder around its major axis. The object model ignores pitch and yaw. Besides, the seabed is assumed to be flat. It is necessary to extend the object modeling to more realistic seabed environments, object shapes, and more complete motions for operational use.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**n**is the unit vector normal to the cylinder surface, and $\kappa $ is the compressive coefficient. Let

**n**be decomposed into

**e**

_{h},

**e**

_{v}) are horizontal and vertical unit vectors (see Figure A1). The sediment compressive normal stress

**F**

_{S}is decomposed as,

**r**is the position vector at any point on the circle and

**r**

_{b}is the position vector at point b with point E as the origin,

**Figure A1.**The location of the axis of rotation of the cylinder in the sediment, b, is determined by the assumption of zero-sum torque to the roll.

_{B}= B/D mildly from near 0.4445 for p

_{B}= 0 and 0.4630. Here, we take λ = 0.453 in this study.

## Appendix B

_{d}) and lift force (F

_{l}) (see Appendix C) roll the object forward with the torque T

_{F}(Figure A2),

_{B}= T

_{w}+ T

_{a},

**Figure A2.**Forces and torques due to drag, lift, buoyancy, and added mass on a partially buried cylinder by combination of ocean currents and bottom wave orbital velocity (U) perpendicular to the major axis of the cylinder.

_{F}> T

_{B}the object accelerates if it is in motion or starts to move if it is at rest (u

_{o}= 0, du

_{o}/dt = 0). When T

_{F}< T

_{B}the object decelerates if it is in motion or keeps motionless if it is at rest. When T

_{F}= T

_{B}the object keeps velocity constant if it is in motion or keeps motionless if it is at rest. Thus, the threshold for the munition’s mobility becomes

_{opb}is the object’s mobility parameter for percentage burial p

_{b}[14] and S

_{o}is the relative density of the object. For motionless munition (u

_{o}= 0), the condition for the object to move is obtained through substituting (A13) into (A12)

_{F}, T

_{B}) in (A10) (A11) into (16) leads to

_{o}is the rolling moment about the symmetric axis of the munition; I

_{A}is the rolling moment of munition about the point b (see Figure A2) using the parallel axis theorem; Π is the volume of the munition.

## Appendix C

_{d}), lift force (F

_{l}), buoyancy force (F

_{w}), and added mass (F

_{a}) exerted on the object for rolling by the perpendicular component, U, are given by

^{2}is the gravitational acceleration; ρ

_{w}= 1025 kg/m

^{3}is the density of seawater; ρ

_{o}is the density of the cylindrical object; (C

_{d}, C

_{l}) are the drag and lift coefficients across-cylinder’s main axis, with vortex shedding caused by the oscillating flow (U) due to waves. If time averaged U within a certain time period being used, the mean coefficients for drag and lift (C

_{d}, C

_{l}), depending solely on the Reynolds number and aspect ratio (see Appendix D), can be used. Since the wave component (i.e., the bottom wave orbital velocity) in

**V**

_{w}for the object model is computed from a linear wave model with the temporal resolution of 30 min, the mean coefficients for drag and lift are used. The vortex shedding from objects is neglected. Besides, when the lift coefficient is less certain, we assume

## Appendix D

_{d}depends on the Reynolds number,

^{−6}m

^{2}/s, is the sea water kinematic viscosity; U is the horizontal water velocity perpendicular to the cylinder’s main axis; and D is the cylinder’s diameter (see Figure 6). For an empirical formula used to calculate C

_{d}[30].

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**Figure 1.**Flow chart of the coupled Delft3D-object model to predict an objects’ mobility and burial.

**Figure 2.**Northern Gulf of Mexico of the coast of Panama City with locations of the shallow quadpod (30°04.81′ N, 85°40.41′ W), deep quadpod (30°03.02′ N, 85°41.34′ W), and the NOAA buoy stations PACF1, PCBF1, and 42039. The study area is enclosed by the red lines. The NOAA buoy station PACF1 is nearest to the study area.

**Figure 4.**Fabricated surrogates, purchased replicas, and fabricated replicas of (

**left**) 155 mm HE M107, (

**middle**) 81 mm mortars, and (

**right**) 25 mm and 20 mm cartridges (from [13]).

**Figure 5.**(

**a**) Locations of deep and shallow quadpods; (

**b**) photo of divers laying the object field during the shallow quadpod deployment; (

**c**) layout of objects laid by divers under the shallow quadpod (from [13]).

**Figure 6.**Study area with bathymetry, depth contours (10 m, 20 m, and 25 m), and computational grids for wave module (red), flow module with coarse resolution (white), and flow module with fine resolution (yellow). The black dot represents the shallow quadpod location, and the white square denotes the deep quadpod location (from [17]).

**Figure 7.**Comparison of Delft3D predictions (red) and observations during TREX13 (black) at the shallow quadpod from 21 April to 7 May 2013: (

**a**) near bed (~0.15 m) longitudinal current v

_{e}(m/s); (

**b**) near bed (~0.15 m) latitudinal current v

_{n}(m/s); (

**c**) water depth h (m); (

**d**) peak period T

_{P}(s); (

**e**) significant wave height H

_{S}(m); (

**f**) computed bottom wave orbital velocity U

_{br}(m/s).

**Figure 8.**Roll of a cylindrical object on the seafloor with large aspect ratio forced by the combination of ocean currents and bottom wave orbital velocity. Here, (π/2 − ϕ) is the angle between

**V**

_{w}and the main axis of the cylinder.

**Figure 9.**Time series of sediment Shields parameter (θ

_{sed}) at the shallow quadpod computed from the Delft3D model output.

**Figure 10.**Model predicted burial percentage p

_{B}(t) with re-exposure parameterization (13) for each object at the shallow quadpod from 20 April to 7 May 2013. The predicted burial percentage p

_{B}(t) is less than 0.5 for all the munitions during the whole time period, except during the storm event from 12:00 5 May to 00:00 6 May 2013. The burial percentage p

_{B}(t) is the same for each object, since it only depends on the sediment characteristics [see Equations (9)–(11)].

**Figure 11.**Model predicted objects’ mobility parameters for percentage burial (θ

_{opb}) at the shallow quadpod from 20 April to 7 May 2013. The red color shows that the condition for rolling an object [θ

_{opb}> 1] is satisfied. The parameters θ

_{opb}is not computed between 12:00 5 May to 00:00 6 May 2013, since the predicted burial percentage p

_{B}(t) is larger than 0.5. Among the eight objects, only A2 and C2 have evident time periods when the condition for rolling an object [θ

_{opb}> 1] is satisfied.

**Figure 12.**Model predicted displacement l(t) for each object at the shallow quadpod from 20 April to 7 May 2013. Among the eight objects, only A2 and C2 were immediately mobile and displaced 20.7 m (A2) and 6.52 m (C2) on 12:00 24 April 2013 (dashed line). Other munitions A5, B5, C4, C6, D3, D6 were completely motionless. According to the TREX13 report [13], the objects A2 and C2 were immediately mobile and transported out of the field of view because they were last seen on 23 April 2013. After 23 April 2013, their locations have never been observed.

**Figure 13.**Positions for all visible objects at the shallow quadpod location up to the maintenance dive performed on 8 May: (

**a**) observation for 20 April–7 May 2013; (

**b**) observation for 13:00–20:00 on 5 May 2013; (

**c**) model prediction for 20 April–7 May 2013; (

**d**) model prediction for 13:00–20:00 on 5 May 2013. Note that Figure 13a,b were copied from [13]. The color bars denote the last time when each object was visible with dates for (

**a**,

**c**) and hour on 5 May for (

**b**,

**d**).

**Table 1.**List of surrogate and replica munitions used during TREX13. A total of 26 objects were deployed and 18 objects were recovered (from [13]). The surrogate munitions were fabricated to have rolling moments within 10% of the estimated rolling moment of the real counterpart.

Type with Diameter | Labels | Materials Type | Recovered | Rolling Moment(10 ^{−4} kg m^{2}) | Volume(10 ^{−5} m^{3}) | Mass(kg) | Density(kg m ^{−3}) |

155 mm, HE, M107 | D5, D6 | Delrin, 304 Stainless Surrogate | D5, D6 | 923.59 | 768.38 | 34.15 | 4444 |

D3, D4 | Aluminum Replica | D3, D4 | 500.48 | 768.38 | 20.91 | 2721 | |

81 mm mortar | C3, C4 | Delrin, 316 Stainless, Aluminum tail fins Surrogate | C3, C4 | 24.73 | 120.93 | 3.76 | 3109 |

C5, C6 | 304 Stainless, Aluminum tail fins Replica | C5, C6 | 50.51 | 120.93 | 8.70 | 7194 | |

C1, C2 | Urethane Replica | 8.34 | 120.93 | 1.45 | 1199 | ||

25 mm cartridge | B5, B6 | Delrin, 316 Stainless Surrogate | B5, B6 | 0.46 | 16.55 | 0.39 | 2356 |

B7, B8 | 304 Stainless Replica | B7, B8 | 1.98 | 16.55 | 1.32 | 7975 | |

B3, B4 | Aluminum Replica | B3, B4 | 0.68 | 16.55 | 0.43 | 2598 | |

B1, B2 | Delrin Replica | 0.35 | 16.55 | 0.23 | 1390 | ||

20 mm cartridge | A5, A6 | Delrin, 316 Stainless Surrogate | A6 | 0.13 | 7.70 | 0.20 | 2597 |

A7, A8 | 304 Stainless Replica | A7 | 0.53 | 7.70 | 0.63 | 8181 | |

A3, A4 | Aluminum Replica | A3, A4 | 0.18 | 7.70 | 0.19 | 2468 | |

A1, A2 | Delrin Replica | 0.09 | 7.70 | 0.11 | 1429 |

**Table 2.**Sediment properties from diver push cores taken during the deployment (D1) and the retrieval (R1) of the instrumentation at the shallow quadpod location (from [13]).

Depth Range (cm) | %Gravel | % Sand | MeanPhi-Value | Standard DeviationPhi-Value | %Porosity | Bulk Density (g/cc) | Void Ratio (e) | |||||||

Core # | D1 | R1 | D1 | R1 | D1 | R1 | D1 | R1 | D1 | R1 | D1 | R1 | D1 | R1 |

0–2 | 0.00 | 0.04 | 100.00 | 99.96 | 2.14 | 2.06 | 0.39 | 0.40 | 38.35 | 39.55 | 2.04 | 2.02 | 0.62 | 0.65 |

2–4 | 0.00 | 0.00 | 100.00 | 100.00 | 2.12 | 2.04 | 0.40 | 0.40 | 39.28 | 40.14 | 2.03 | 2.02 | 0.65 | 0.67 |

4–6 | 0.00 | 0.02 | 100.00 | 99.98 | 2.13 | 2.08 | 0.42 | 0.46 | 39.13 | 38.96 | 2.03 | 2.03 | 0.64 | 0.64 |

6–8 | 0.02 | 0.01 | 99.98 | 99.99 | 2.23 | 2.21 | 0.43 | 0.44 | 38.84 | 39.46 | 2.04 | 2.03 | 0.63 | 0.65 |

8–10 | 0.13 | 0.01 | 99.87 | 99.99 | 1.94 | 2.24 | 0.62 | 0.40 | 37.62 | 39.26 | 2.06 | 2.03 | 0.60 | 0.65 |

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**MDPI and ACS Style**

Chu, P.C.; Pessanha, V.S.; Fan, C.; Calantoni, J.
Coupled Delft3D-Object Model to Predict Mobility of Munition on Sandy Seafloor. *Fluids* **2021**, *6*, 330.
https://doi.org/10.3390/fluids6090330

**AMA Style**

Chu PC, Pessanha VS, Fan C, Calantoni J.
Coupled Delft3D-Object Model to Predict Mobility of Munition on Sandy Seafloor. *Fluids*. 2021; 6(9):330.
https://doi.org/10.3390/fluids6090330

**Chicago/Turabian Style**

Chu, Peter C., Vinicius S. Pessanha, Chenwu Fan, and Joseph Calantoni.
2021. "Coupled Delft3D-Object Model to Predict Mobility of Munition on Sandy Seafloor" *Fluids* 6, no. 9: 330.
https://doi.org/10.3390/fluids6090330