Sediment transport analysis under combined action of waves and current using a novel semi-coupled computational fluid dynamics solver

Abstract

Development of a numerical model to study the hydrodynamics and sediment transport phenomenon is essential in venturing into the field of coastal engineering applications. In the framework of OpenFOAM, a solver which can simulate simultaneously waves and current and its impact on sediment transport is nonexistent. Here, a semi-coupled sediment transport solver has been developed using open source framework OpenFOAM. Initially a new hydrodynamics solver (hereafter named hydroFOAM) has been developed to study the flow dynamics in coastal engineering applications. A set of Navier–Stokes equations including the continuity, momentum and inter-phase equations are solved using the finite volume method. Volume of fluid method has been employed to track the free surface. Later, different mathematical formulations are solved in a newly written solver (hereafter named sedimentTransportFOAM) to simulate the sediment transport phenomenon. Bed load and suspended load transport theories are being used while developing the solver. Precise descriptions of the two solver as well as the employed algorithm has been presented in this paper. Towards the end, the developed mathematical model has been validated with earlier experimental investigations and same has been presented using graphical figures and tables.

Appendices

Appendix A: MULES

The multidimensional universal limiter for explicit solution (MULES) technique as described in Sect. 2.1 is presented here. The Eq. (5) can be written in the following form:

$$\begin{aligned} \frac{\partial A}{\partial t}+\nabla \cdot \textbf{B}=0 \end{aligned}$$

(28)

The Eq. (28) can be discretized and written in the following form:

$$\begin{aligned} \frac{A^{n+1}_{i} - A^{n}_{i}}{\Delta t}V+\sum _f\left( \mathbf {B^n \cdot S}\right) _f=0 \end{aligned}$$

(29)

The Eq. (29) can be expressed with flux-corrected transport schemes. A sample one-dimensional simple flux-corrected transport scheme is presented below for the reference of the reader.

$$\begin{aligned} A_i^{n+1} = A_{i}^{n}-\frac{\Delta t}{V}\left( B_{i+\frac{1}{2}}^{L}-B_{i-\frac{1}{2}}^{L}\right) -\frac{\Delta t}{V}\left( \lambda _{i+\frac{1}{2}}C_{i+\frac{1}{2}}-\lambda _{i-\frac{1}{2}}C_{i-\frac{1}{2}}\right) \end{aligned}$$

(30)

A sample MULES function used in OpenFOAM is presented below:

MULES::correct

(

   alpha1,

   alphaPhi,

   talphaPhiCorr0.ref(),

   1,

   0

)

Appendix B: Turbulence modeling

The OpenFOAM framework has three different types of trbulene modeling provided with the package which are being rendered in the hydroFOAM and sedimentTransportFoam solver. These are Reynolds averaged simulation (RAS), detached eddy simulation (DES), and large eddy simulation (LES). The details of the implementation of all these models are prsented in the OpenFOAM documentation (https://www.https://openfoam.org/). $k-\epsilon$ turbulence model is being presented here for the reference of readers. Two transport-equation for linear-eddy-viscosity turbulence closure model are as follows:

$$\begin{aligned} \frac{D}{Dt}\left( \rho k\right) = \nabla \cdot \left( \rho D_k\nabla k\right) +P-\rho \epsilon \end{aligned}$$

(31)

where $D_k$ is effective diffusivity of k, P is turbulent kinetic energy production rate, $\epsilon$ is turbulent kinetic energy dissipation energy rate.

$$\begin{aligned} \frac{D}{Dt}\left( \rho \epsilon \right) = \nabla \cdot \left( \rho D_\epsilon \nabla \epsilon \right) +\frac{C_l\epsilon }{k}\left( P+C_3\frac{2}{3}k\nabla \cdot \textbf{u}\right) -C_2\rho \frac{\epsilon ^2}{k} \end{aligned}$$

(32)

where $D_\epsilon$ is effective diffusivity for $\epsilon$ and $C_1$, $C_2$, and $C_3$ are model coefficients.

The above Eqs. (31) and (32) are solved using numerical methods. The $k-\epsilon$ model is extensively used to predict the turbulence modeling influence in the flow domain.