# Oil Droplet Transport under Non-Breaking Waves: An Eulerian RANS Approach Combined with a Lagrangian Particle Dispersion Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Governing Equations: Eulerian RANS Framework

_{t}is eddy viscosity. Note that the eddy viscosity tensor is taken to be diagonal with equal diagonal entries, thus we consider an isotropic eddy diffusivity. The k-ε model isotropic eddy viscosity is given by:

#### 2.2. Governing Equations: Lagrangian Particle Dispersion Framework

^{3}, and considering the largest droplet sizes herein, which is 1000 microns, the term ${\tau}_{\rho}$ is around $0.05s$. This gives a Stokes number ${S}_{t}<$0.05, which is a relatively small value (note that S

_{t}= 0 for neutrally buoyant particles) indicating that inertial effects can be neglected.

#### 2.3. Problem Setup

#### 2.3.1. Wave Formulation

^{3}, air density as $1.225$ kg/m

^{3}, water dynamic viscosity as $1.003\times {10}^{-3}$ kg/m

^{3}, air dynamic viscosity as $1.7894\times {10}^{-5}$ kg/m

^{3}, and oil density as $866$ kg/m

^{3}, representing Alaskan North Slope oil [36].

#### 2.3.2. Flow Simulation

#### 2.3.3. Particle Tracking

_{w}. The NEMO3D code is capable of constructing a triangular unstructured mesh over any set of points and using linear interpolation to calculate the in-between values of the variables. The particle search algorithm locates the particle and links it to the corresponding triangular element in which the particle has traveled to. Moreover, the transient input data are updated at each tracking time step. Using an unstructured linear mesh provided the capability of constructing the spatial gradients of eddy diffusivity.

## 3. Results

#### 3.1. Numerical Simulation Validation

#### 3.2. Vertical Profiles of Turbulent Quantities

#### 3.3. Particle Trajectories

#### 3.4. Effect of Turbulent Diffusion

_{2}= 0.80 m. It is apparent that the movement is dominated by buoyancy, which is due to the large buoyancy of the droplets. Particles initially travel upward until reaching an equilibrium depth modulated by the action of the wave motion and the turbulence by the end of the simulation. The trajectory tends to be more horizontal at x

_{2}≈ 1.05 m corresponding to the region of highest diffusivity, which is consistent with the slowing down of 100 μm droplets as they travel downward. It is also consistent with the results mentioned earlier [17].

#### 3.5. Stokes Drift Calculations

^{−1}while the theoretical value [27] was $0.075$ m s

^{−1}. The fact that the two values are close suggests that the RANS simulation is capturing, at least, second-order accurate kinematics.

^{−1}and $0.062$ m s

^{−1}, respectively. The smaller Stokes velocity compared to the case without turbulence effect is because turbulence (i.e., randomness) results in a net downward movement of the plume (discussed earlier), where the Stokes velocity is smaller. The gradient of eddy diffusivity in this case appears to have only a slight impact on the Stokes drift, increasing the velocity by 5% (from $0.062$ to $0.065$ m s

^{−1}). The greater Stokes drift induced by the gradient of eddy diffusivity is consistent with the fact that the plume remains closer to the surface when the gradient is included (Figure 10).

## 4. Discussion

_{2}≈ 1.1 m causes an upward advective velocity. Therefore, accounting for the turbulent diffusivity slows down the predominantly downward movement of neutral (and low) buoyancy droplets (or particles).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Details of multi-phase wave simulation. The areas colored in red represent cells with water phase (i.e., α

_{w}= 1 in Equation (7a)) and areas colored in blue represent cells with air phase (i.e., α

_{w}= 0).

**Figure 2.**Details of the problem domain (

**top**) and mesh refinement near the surface (

**bottom**). The darker blue represents areas with more refined mesh.

**Figure 3.**Instantaneous contours of vertical (

**top**) and horizontal (

**bottom**) velocities after the flow is fully developed. Horizontal velocity is positive to the right and vertical velocity is positive upward.

**Figure 4.**Comparison of the numerical solution with the analytical solution in terms of time series of horizontal (

**a**) and vertical (

**b**) velocities at the point corresponding to x

_{1}= 2.7 m and x

_{2}= 1.1 m during 10 periods of the wave for the fully developed flow.

**Figure 5.**Comparison of horizontal (

**a**) and vertical (

**b**) velocities between numerical (solid line) and analytical solutions at x

_{1}= 2.7 m at two different times corresponding to the crest and trough of a wave. In the left panel, the left curves are under the trough while the right curves are under the crest. In the right panel, the left curves occurred under the crest while the right curves occurred under the trough.

**Figure 6.**Vertical variation of turbulent kinetic energy (m

^{2}s

^{−2}) (

**a**), turbulence kinetic energy (TKE) dissipation rate (m

^{2}s

^{−3})

**(b)**, and eddy diffusivity (m

^{2}s

^{−1}) (

**c**) underneath a crest (x

_{1}= 1.56 m) and a trough (x

_{1}= 2.34 m) of the wave.

**Figure 7.**Particle positions at different times. Particles of diameter 1000 μm are represented with green points and the red points represent particles of diameter equal to 100 μm. Each group has 50 particles that are tracked during 10 wave periods. Starting point of all particles was at ${x}_{2}=1.10$ m.

**Figure 8.**Ensemble averaged plume trajectory (over 500 particles of each size) of particles of diameter 100 and 1000 μm. The particles were released at ${x}_{1}\in [2.5,\text{}3]$ m, ${x}_{2}=1.10$ m. The dotted line through the plume trajectory was obtained by window averaging over each trajectory loop corresponding to the wave period.

**Figure 9.**Ensemble averaged plume trajectory of particles of diameters 100 and 1000 μm (500 particles were used to obtain the averages). The particles were released at ${x}_{1}\in [2.5,\text{}3]$ m, ${x}_{2}=0.8$ m. Dotted line through the plume trajectory was obtained by window averaging over each trajectory loop corresponding to the wave period.

**Figure 10.**Ensemble averaged plume trajectory (over 500 particles of each size) of particles of diameter 100 and 1000 μm. Release location was at ${x}_{1}\in [2.5,\text{}3]$ m, ${x}_{2}=1.1$ m. Dotted line through the plume trajectory was obtained by window averaging over each trajectory loop corresponding to the wave period.

**Figure 11.**Ensemble averaged plume trajectory (over 500 neutrally buoyant particles of each size) for the case with no turbulence effect and thus with pure advection. Release location was at ${x}_{1}\in [2.5,\text{}3]$ m, ${x}_{2}=1.10$ m.

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**MDPI and ACS Style**

Golshan, R.; Boufadel, M.C.; Rodriguez, V.A.; Geng, X.; Gao, F.; King, T.; Robinson, B.; Tejada-Martínez, A.E.
Oil Droplet Transport under Non-Breaking Waves: An Eulerian RANS Approach Combined with a Lagrangian Particle Dispersion Model. *J. Mar. Sci. Eng.* **2018**, *6*, 7.
https://doi.org/10.3390/jmse6010007

**AMA Style**

Golshan R, Boufadel MC, Rodriguez VA, Geng X, Gao F, King T, Robinson B, Tejada-Martínez AE.
Oil Droplet Transport under Non-Breaking Waves: An Eulerian RANS Approach Combined with a Lagrangian Particle Dispersion Model. *Journal of Marine Science and Engineering*. 2018; 6(1):7.
https://doi.org/10.3390/jmse6010007

**Chicago/Turabian Style**

Golshan, Roozbeh, Michel C. Boufadel, Victor A. Rodriguez, Xiaolong Geng, Feng Gao, Thomas King, Brian Robinson, and Andrés E. Tejada-Martínez.
2018. "Oil Droplet Transport under Non-Breaking Waves: An Eulerian RANS Approach Combined with a Lagrangian Particle Dispersion Model" *Journal of Marine Science and Engineering* 6, no. 1: 7.
https://doi.org/10.3390/jmse6010007