### Abstract

Direct numerical simulation and large-eddy simulation are developed to investigate waterwaves propagating over viscous fluid mud at the bottom, with a focus on the study of wave breaking case. In the simulations, the watersurface and the water–mud interface are captured with a coupled level-set and volume-of-fluid method. For non-breaking waterwaves of finite amplitude, it is found that the overall wave decay rate is in agreement with the existing linear theory. For breaking waterwaves, detailed description of the instantaneous flow field is obtained from the simulation. The time history of the total mechanical energy in water and mud shows that during the early stage of the wave breaking, the energy decays slowly; then, the energy decays rapidly; and finally, the decay rate of energy becomes small again. Statistics of the total mechanical energy indicates that the mud layer reduces the wave breaking intensity and shortens the breaking duration significantly. The effect of mud on the energy dissipation also induces a large amount of energy left in the system after the wave breaking. To obtain a better understanding of the underlying mechanism, energy transport in water and mud is analyzed in detail. A study is then performed on the viscous dissipation and the energy transfer at the water–mud interface. It is found that during the wave breaking, the majority of energy is lost at the watersurface as well as through the viscous dissipation in mud. The energy and viscous dissipation in mud and the energy transfer at the water–mud interface are strongly affected by the wave breaking at the watersurface.

Y.H. and X.L. are supported by the fellowship provided by the China Scholarship Council for their studies at the Johns Hopkins University. X.G., Y.L., R.A.D., and L.S. are supported by the Office of Naval Research through the MURI project “Mechanisms of Fluid-Mud Interactions under Waves” (Grant No. N00014-06-1-0718). The simulations were performed on the supercomputers provided by the High Performance Computing Modernization Program (HPCMP) of the Department of Defence.

I. INTRODUCTION

II. PROBLEM DEFINITION AND NUMERICAL APPROACH

A. Canonical problem and mathematical formulation

B. Numerical scheme

C. Numerical treatment of free surface

D. Simulation parameters

E. Comparison of non-breaking waterwave result with previous analytical solution

III. RESULTS OF BREAKING WATERWAVES

A. Breaking process and energy evolution

B. Transport of kinetic energy

C. Viscous dissipation and energy transfer between water and mud

IV. CONCLUSIONS

### Key Topics

- Viscosity
- 64.0
- Surface waves
- 22.0
- Turbulent flows
- 21.0
- Hydrodynamic waves
- 19.0
- Energy transfer
- 18.0

## Figures

Schematics of water wave over a mud layer.

Schematics of water wave over a mud layer.

Time history of turbulent velocity fluctuation in the three-dimensional simulation of case NBW-B1 with initial turbulence seed added.

Time history of turbulent velocity fluctuation in the three-dimensional simulation of case NBW-B1 with initial turbulence seed added.

Time history of (a) the amplitude of the surface wave *a* normalized by its initial value *a* _{0} and (b) the ratio of interfacial wave amplitude to surface wave amplitude *b*/*a*. Prediction of DL’s theory for case NBW-B0 (——), case NBW-B1 (– – –), and case NBW-B2 (– · – · –). DNS results for case NBW-A1 (△), case NBW-B0 (▽), case NBW-B1 (○), case NBW-B2 (◊), and case NBW-C1 (•).

Time history of (a) the amplitude of the surface wave *a* normalized by its initial value *a* _{0} and (b) the ratio of interfacial wave amplitude to surface wave amplitude *b*/*a*. Prediction of DL’s theory for case NBW-B0 (——), case NBW-B1 (– – –), and case NBW-B2 (– · – · –). DNS results for case NBW-A1 (△), case NBW-B0 (▽), case NBW-B1 (○), case NBW-B2 (◊), and case NBW-C1 (•).

Instantaneous flow fields on the *x*–*y* cross section at the center of the computational domain (i.e., *z* = 0) with contours of *u* at (a) *t* = 0.78*T*, (b) *t* = 0.99*T*, (c) *t* = 1.53*T*, and (d) *t* = 2.25*T*. The distribution of ε in the mud layer is also shown. Here, *u* is normalized by *a* _{0}σ and ε is normalized by . The water–mud interface is marked by (– – –). The arrow at the water surface denotes the wave propagation direction.

Instantaneous flow fields on the *x*–*y* cross section at the center of the computational domain (i.e., *z* = 0) with contours of *u* at (a) *t* = 0.78*T*, (b) *t* = 0.99*T*, (c) *t* = 1.53*T*, and (d) *t* = 2.25*T*. The distribution of ε in the mud layer is also shown. Here, *u* is normalized by *a* _{0}σ and ε is normalized by . The water–mud interface is marked by (– – –). The arrow at the water surface denotes the wave propagation direction.

Time history of (a) total mechanical energy in water and mud, (b) total mechanical energy in water, and (c) total mechanical energy in mud of case BW-A1 (——), case BW-A2 (– – –), case BW-B1 (– · – · –), and case BW-C1 (· · · · · ·). (a) The theoretical prediction for non-breaking water waves is also shown (with symbol).

Time history of (a) total mechanical energy in water and mud, (b) total mechanical energy in water, and (c) total mechanical energy in mud of case BW-A1 (——), case BW-A2 (– – –), case BW-B1 (– · – · –), and case BW-C1 (· · · · · ·). (a) The theoretical prediction for non-breaking water waves is also shown (with symbol).

Vertical transport terms in the horizontally-averaged kinetic energy budget equation in the water region for (a) non-breaking wave and (b) breaking wave: (——), transport due to advection ; (– – –), transport due to pressure ; (– · · – · · –), transport due to viscous diffusion . The results are normalized by *E* ^{ va }σ. Here, the superscript “*va*” denotes the volume-averaged value. For the non-breaking wave, case NBW-B1 at *ak* = 0.15 is shown. For the breaking wave, case BW-A1 at *t* = 2.25*T* is shown. Note that the scales are different between the two figures.

Vertical transport terms in the horizontally-averaged kinetic energy budget equation in the water region for (a) non-breaking wave and (b) breaking wave: (——), transport due to advection ; (– – –), transport due to pressure ; (– · · – · · –), transport due to viscous diffusion . The results are normalized by *E* ^{ va }σ. Here, the superscript “*va*” denotes the volume-averaged value. For the non-breaking wave, case NBW-B1 at *ak* = 0.15 is shown. For the breaking wave, case BW-A1 at *t* = 2.25*T* is shown. Note that the scales are different between the two figures.

Vertical transport terms in the horizontally-averaged kinetic energy budget equation in the mud region for (a) non-breaking wave and (b) breaking wave: (——), transport due to advection ; (– – –), transport due to pressure ; (– · · – · · –), transport due to viscous diffusion ; (· · · · · ·), viscous dissipation . The results are normalized by *E* ^{ va }σ. For the non-breaking wave, case NBW-B1 at *ak* = 0.15 is shown. For the breaking wave, case BW-A1 at *t* = 2.25*T* is shown.

Vertical transport terms in the horizontally-averaged kinetic energy budget equation in the mud region for (a) non-breaking wave and (b) breaking wave: (——), transport due to advection ; (– – –), transport due to pressure ; (– · · – · · –), transport due to viscous diffusion ; (· · · · · ·), viscous dissipation . The results are normalized by *E* ^{ va }σ. For the non-breaking wave, case NBW-B1 at *ak* = 0.15 is shown. For the breaking wave, case BW-A1 at *t* = 2.25*T* is shown.

Time history of (a) the viscous dissipation in water, (b) the viscous dissipation in mud, (c) the work done at the water–mud interface, (d) the work done by pressure at the water–mud interface, and (e) the ratio of kinetic energy between mud and water for case BW-A1 (——), case BW-A2 (– – –), case BW-B1 (– · – · –), and case BW-C1 (· · · · · ·).

Time history of (a) the viscous dissipation in water, (b) the viscous dissipation in mud, (c) the work done at the water–mud interface, (d) the work done by pressure at the water–mud interface, and (e) the ratio of kinetic energy between mud and water for case BW-A1 (——), case BW-A2 (– – –), case BW-B1 (– · – · –), and case BW-C1 (· · · · · ·).

## Tables

Parameters considered in the present study. The “NBW” stands for non-breaking water wave; “BW” stands for breaking water wave. The *a* _{0} is the initial wave amplitude.

Parameters considered in the present study. The “NBW” stands for non-breaking water wave; “BW” stands for breaking water wave. The *a* _{0} is the initial wave amplitude.

Energy dissipated through the viscous dissipation in water and mud, the energy transfer at water–mud and air–water interfaces, and the energy loss during the wave breaking.

Energy dissipated through the viscous dissipation in water and mud, the energy transfer at water–mud and air–water interfaces, and the energy loss during the wave breaking.

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