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2. Governing Fluid Mechanics

To describe water waves mathematically, we begin with the foundational equations of fluid mechanics, specifically the Navier-Stokes equations. These equations govern the behavior of moving fluids by relating pressure, velocity, temperature, and density, based on conservation principles. The Navier-Stokes equations encompass conservation of mass, momentum, and energy, and as we proceed, we will apply certain simplifying assumptions to make these equations more manageable for describing water waves.

For wind-driven gravity waves, the density of water remains approximately constant over space and time, allowing us to use the incompressible Navier-Stokes equations.

Conservation of Mass

The continuity equation (or conservation of mass) for a fluid is given by:

\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \boldsymbol{u}) = 0

where \rho is density, t is time, and u is the velocity flow field (in one, two, or three dimensions). Assuming that density is constant (incompressible flow), the conservation of mass statement simplifies to:

\nabla \cdot \boldsymbol{u} = 0

This simplification allows us to focus on changes in the velocity field without needing to account for density variations.

Conservation of Momentum

The conservation of momentum equation is given by

(\frac{\partial}{\partial t} + \boldsymbol{u} \cdot \nabla) \boldsymbol{u} = -\nabla (\frac{p}{\rho} + g z) + \nu \nabla^2 \boldsymbol{u}

where \nu is the kinematic viscosity, defined as the ratio of the dynamic viscosity \mu over the fluid density \rho ( \nu = \frac{\mu}{\rho}). The term gz represents the gravitational potential energy per unit mass.

Vorticity

Vorticity of the flow, \boldsymbol{\Omega} , describes the local spinning motion of the fluid and is obtained by taking the curl of the velocity field:

\boldsymbol{\Omega} = \nabla \times \boldsymbol{u}

The evolution of vorticity can be expressed by the following equation:

(\frac{\partial}{\partial t} + \boldsymbol{u} \cdot \nabla) \boldsymbol{\Omega} = \boldsymbol{\Omega} \cdot \nabla \boldsymbol{u} + \nu \nabla^2 \boldsymbol{u}

The viscosity of water is small enough that we can neglect the final term, resulting in the simplified expression,

(\frac{\partial}{\partial t} + \boldsymbol{u} \cdot \nabla) \boldsymbol{\Omega} = \boldsymbol{\Omega} \cdot \nabla \boldsymbol{u}

In cases where the flow is irrotational, we use Laplace’s equation (from Chapter 6), which states that the velocity potential \phi satisfies:

\nabla^2 \phi = 0

The Momentum Equation in Terms of Velocity Potential

When the flow is irrotational, we can reframe the momentum equation using the velocity potential \phi, leading to

\nabla [ \frac{\partial \phi}{\partial t} + \frac{1}{2} | \nabla \phi |^2 ] = \nabla(\frac{p}{\rho} + gz)

Finally, integrating the momentum equation with respect to space variables (x, y, z), and with slight rearrangement, yields the unsteady Bernoulli equation,

\frac{p}{\rho} =gz + \frac{\partial \phi}{\partial t} + \frac{1}{2} | \nabla \phi |^2 + c(t)

Here c(t) is a constant of integration that is a function of time, since we integrated with respect to space. You may recall from your fluid mechanics course that Bernoulli’s equation is another form of the conservation of energy, that includes contributions from pressure, kinetic energy, and potential energy. In this formulation, the kinetic energy contribution can be seen in the \frac{1}{2} | \nabla \phi |^2 term, while potential energy can be seen in the hydrostatics term gz.

With the governing equations in place, we next require initial and boundary conditions to fully define the problem and complete the system of equations. These conditions will vary depending on the wave and flow characteristics, and they will be the focus of the next section.

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Ocean Hydrodynamics for Engineers Copyright © by Maija Benitz is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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