# 3.1.4. Ravel et al. (2009) “Numerical Simulation of Viscous Nonlinear Waves”¶

Ravel, A., Wen, X., Smith, M.H., “Numerical Simulation of Viscous, Nonlinear and Progressive Water Waves”, Jounral of Fluid Mechanics, 637 (2009) pp 443 - 473

## 3.1.4.1. Where is this study placed within the wave modelling community?¶

This study is in the category of models attempting to solve the problem of non-linear wave effects in shallow water.

It takes the approach of deterministic (phase resolving) wave modelling in the time domain.

It uses one phase per realisation, and seven realisations are performed (which may have been limited by the large computing time needed) but at different depths and frequencies.

Unlike other models in this category, it also includes viscous effects, which allows energy decay rate to be observed, which is one of the open questions in this field.

It also postulates various mechanisms for this decay rate, to try to explain the physics behind the energy decay rate difference between intermediate and deep water cases.

## 3.1.4.2. Summary of Findings¶

Near the water surface:

Clockwise and anticlockwise rotation of the fluid at the trough and crest

Thicker vorticity layers and larger magnitude of vorticity compared with:

Inviscid, rotational flow (the Gerstner wave)

Low Re viscous flow (from Kinsman (1965) and Lamb (1932))

Thick shear layer with larger shear stress (-ve and +ve shear stress below crest and troughs of the wave)

Good comparison between wave energy decay rates (numerical, experimental and theoretical)

## 3.1.4.3. Literature Review¶

3 categories of techniques:

Potential flow - inviscid, irrotational, no turbulence (analytical solution via Laplace equation for velocity potential plus boundary conditions):

Inaccurate for flows with a boundary layer (e.g. atmospheric boundary layer, bottom boundary layer)

Vorticity is zero, so fluid loses it’s infinite degrees of freedom (so the fluid ceases to be a fluid in reality)

Inviscid rotational flow:

Gersther’s trochodial wave theory

Boundary integral method - periodic and solitary waves with constant vorticity - limited due to special application of vorticity distribution

Boundary integral method - interaction of small amplitude waves with bottom ripples

Viscous flow:

Kinsman and Lamb (low Re): No convection terms, which linearises the Navier Stokes Equations, such that an analytical solution can be found

Behroozi: conservation of energy for fluid viscosity and decay relation

Wang & Joseph: viscous potential flow and viscous correction potential flow for decay of free gravity wave due to viscosity

Dutykh & Dias: Visco-potential free surface flow

### 3.1.4.3.1. What kind of numerical methods are there?¶

Front tracking method

Boundary integral method

Phase field method

Second gradient method

Level set method

Marker and cell method

Smooth particle hydrodynamics

Finite analytic method and modified marker and cell method, using zero shear stress at the surface for velocity and energy decay rate

Kinematic boundary condition at the surface - zero shear stress, zero pressure or total zero shear stress on water surface - no air considered only water

Volume of fluid method - Hirt and Nichols (1981) for shape of free surface. Zwart, Burns and Galpin (2007) - VOF - define volume fraction of air and water separately with high resolution scheme for mass conservation in air and water, v, p and aw, af solved simultaneously in coupled system. Advantage of VOF = includes air and water to retain mass continuity and eliminates need to reconstruct the shape of the water surface during the calculation, like MAC, SPH and height function methods

### 3.1.4.3.2. What phenomena are we testing for?¶

Waves are transient, non-linear, rotational and viscous and often simplified to linear, irrotational and inviscid - this removes effect of vorticity and shear stress, which are strong indicators of rotational and viscous behaviour.

Past vorticity work - velocity as a function of vorticity via Biot-Savart integral, eliminating pressure from the formulation. Zero shear stress - no wind at surface.

### 3.1.4.3.3. Do we need vorticity?¶

Yes - parasitic capillaries cause it - orbital vorticity below waves with wind - causes Langmuir circulations or turbulence balance - strong consequences. Vorticity is proportional to gradient of velocity - applicable for high Re flows. Action of viscous force shown in high vorticity and shear stress.

### 3.1.4.3.4. Questions arising from literature review¶

Is there vorticity and shear stress in a progressive wave without wind? (air following water)

If so, what is the maximum value of vorticity and shear stress in the water? Are they symmetric under crest and trough?

Is there an effect of water depth on vorticity and shear stress?

## 3.1.4.4. Method¶

### 3.1.4.4.1. Assumptions¶

2D

Viscous fluid

Non-linear

Non-breaking

Intermediate and deep water - with a slope

Zero wind velocity - air follows wind

**Laminar flow**

### 3.1.4.4.2. Inputs¶

\(L\) = wavelength

\(T\) = periodic time

\(a\) = wave amplitude

\(k = 2 \pi / L\) = wavenumber

\(\sigma = 2 \pi / T\) = angular frequency

\(c = L / T = \sigma / k\) = angular frequency

\(2a / L\) = wave steepness

\(h\) = depth in water

\(h'\) = depth in air

### 3.1.4.4.3. Outputs¶

Velocity field

Streamlines

Vorticity

Shear stress

### 3.1.4.4.4. Physics¶

**Boundary Conditions are:**

Walls top and bottom

Opening at inlet and outlet - non-linear, inviscid flow solution at inlet

A 1:15 slope to dissipate the wave energy on the beech

**2D Equations:**

Continuity equation for water

Continuity equation for air

Momentum equation for mixture in x

Momentum equation for mixture in y

Pressure-velocity coupling in x

Pressure-velocity coupling in y

**Unknowns:**

Volume fraction of water (scalar) (x,y) - by volume fraction constraint, we get the volume fraction of air (x,y)

Velocity of water (x)

Velocity of water (y)

Velocity of air (x)

Velocity of air (y)

Pressure of mixture (scalar) (x,y)

**Initial Conditions:**

Non-linear potential flow solution

**Correction for outward flow:**

Asymmetry of flow means more flow moves under wave crest than out under trough

Hence, slope is moved away at a velocity found by integrating the potential flow solution

### 3.1.4.4.5. Numerical Discretisation¶

Continuity Equation:

Second order backward Euler for transient term

First/second order blended scheme for advection term

Momentum Equation:

Second order backward Euler for transient term

Second order scheme for convection term, diffusion term and pressure gradient term

Pressure-velocity coupling:

Rhie-Chow interpolation

### 3.1.4.4.6. Grid Discretisation¶

Structured mesh

16 grid points over wave height

100 grid points over wavelength

### 3.1.4.4.7. Solver¶

Algebraic multi-grid method

### 3.1.4.4.8. Test Cases¶

All these waves are:

Non-linear, i.e. \(ka > 0.1\)

Non-breaking, i.e. \(2a/L \le 0.08\)

Case |
2a/L (steepness) |
h/L (depth ratio) |
T (period) |
Description |
---|---|---|---|---|

1 (C1) |
0.04 |
0.2 |
0.7592 |
not steep, intermediate depth |

2 (C2) |
0.04 |
0.6 |
0.6 |
not steep, deep |

3 (C3) |
0.06 |
0.2 |
0.7592 |
steep, intermediate depth |

4 (C4) |
0.06 |
0.6 |
0.6 |
steep, deep |

5 (C5) |
0.08 |
0.2 |
0.7592 |
very steep, intermediate depth |

6 (C6) |
0.08 |
0.6 |
0.6 |
very steep, deep |

7 (CEX) |
0.06 |
0.44 |
0.7 |
for comparison with experiment and energy density |

## 3.1.4.5. Results¶

### 3.1.4.5.1. Energy Decay Rate¶

Determine Kinetic, Potential and Total Energy Decay Rate

Compute decay rate \(\beta\) by fitting this equation:

Decay rate is higher for deep water case than for intermediate water case

**The rest of the paper discusses why by deep water has a higher energy decay rate by attempting to describe the physics**

### 3.1.4.5.2. Free Surface Profiles¶

Comparing the free surface profiles of:

Deep water case

Intermediate depth water case

Nonlinear Stokes wave

Shows:

Deep water case is

**Symmetric**Intermediate depth water case is

**Asymmetric**

Hence there is an effect of the bottom surface, forcing water to move towards the crest in intermediate depths

**Why does the fluid move upwards towards the crest in the case of intermediate depth?**

### 3.1.4.5.3. Velocity field¶

Intermediate water case shows higher peak velocity in the crest than the deep water case.

### 3.1.4.5.4. Streamlines¶

Fluid moves to where it encounters least resistance, this point is higher for intermediate water than deep water.

### 3.1.4.5.5. Vorticity¶

Vorticity distribution is oscillatory with depth

The oscillations are caused by viscosity

The maximum vorticity is larger and has a thicker layer for intermediate compared with deep cases

However, the deep depth has more oscillations in the vorticity field, because it is deeper

### 3.1.4.5.6. Shear Stress¶

Intermediate depth case also has larger shear stress than deep case, because the bottom boundary restricts motion of the water, providing larger velocity gradient, hence larger shear stress.

The deep water case has a bottom boundary located deeper, so the fluid at the surface is less influenced by viscous effects

### 3.1.4.5.7. Effect of Wave Steepness¶

Wave steepness also increases vorticity and shear stress because the velocity gradients are higher.

**The effect of the oscillatory vorticity field with depth seems to be a fundamental phenomenon and affects the shear stress and velocity field**

**The interface between the air and water is the most critical region in the flow, due to the high viscous shear stress created there**