Conservation Laws and Boundary Conditions

The Ocean Environment

Non Linear Free-surface Condition

[math]\displaystyle{ \begin{matrix} &\bullet(X,Y,Z) &: &\mbox{Earth Fixed Coordinate System} \\ &\vec X &: &\mbox{Fixed Eulerian Vector} \\ &\vec V &: &\mbox{Flow Velocity Vector at} \ \vec X \\ &\zeta &: &\mbox{Free Surface Elevation} \end{matrix} }[/math]

[math]\displaystyle{ \bullet }[/math] Assume ideal fluid (No shear stresses) and irrotational flow:

[math]\displaystyle{ \nabla \times \vec V = 0 }[/math]

Let:

[math]\displaystyle{ \vec V = \nabla \Phi \Rightarrow \nabla \times \nabla \Phi \equiv 0 }[/math]

Where [math]\displaystyle{ \Phi(\vec{X},t) }[/math] is the velocity potential assumed sufficiently continuously differentiable.

[math]\displaystyle{ \bullet }[/math] Potential flow model of surface wave propagation and wave-body interactions very accurate. Few important exceptions will be noted.

[math]\displaystyle{ \bullet }[/math] Conservation of mass:

[math]\displaystyle{ \nabla \cdot \vec V = 0 \Rightarrow }[/math]

[math]\displaystyle{ \nabla \cdot \nabla \Phi = 0 \Rightarrow \nabla^2 \Phi = 0 }[/math]

or

[math]\displaystyle{ \frac{\partial^2 \Phi}{\partial X^2} + \frac{\partial^2\Phi}{\partial Y^2} + \frac{\partial^2\Phi}{\partial Z^2} = 0, \quad \mbox{Laplace Equation} }[/math]

[math]\displaystyle{ \bullet }[/math] Conservation of Linear momentum. Euler's Equation in the Absence of Viscosity.

[math]\displaystyle{ \frac{\partial \vec V}{\partial t} + (\vec V \cdot \nabla) \vec V = - \frac1{\rho} \nabla P + \vec g }[/math]

[math]\displaystyle{ P(\vec X, t) : \mbox{Fluid Pressure at} (\vec X, t) }[/math]

[math]\displaystyle{ \vec g = - \vec k g : \mbox{Acceleration of Gravity} }[/math]

[math]\displaystyle{ \vec k : \mbox{unit vector pointing in the positive z-direction} }[/math]

[math]\displaystyle{ \rho : \mbox{water density} \, }[/math]

[math]\displaystyle{ \bullet }[/math] Vector Identity:

[math]\displaystyle{ (\vec V \cdot \nabla) \vec V = \frac 1{2} \nabla (\vec V \cdot \vec V) - \vec V \times ( \nabla \times \vec V) }[/math]

in irrotational flow: [math]\displaystyle{ \nabla \times \vec V = 0 }[/math], thus Euler's equations become:

[math]\displaystyle{ \frac{\partial \vec V}{\partial t} + \frac 1{2} \nabla (\vec V \cdot \vec V) = - \frac 1{\rho} \nabla P - \nabla (g Z) }[/math]

[math]\displaystyle{ \mbox{Note} : \quad \nabla Z = \vec K, \vec V = \nabla \Phi }[/math]

Upon substitution:

[math]\displaystyle{ \nabla \underbrace{(\frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{e} + g Z )} = 0 }[/math]

[math]\displaystyle{ F ( \vec X, t) }[/math] [math]\displaystyle{ \nabla F( \vec X, t) =0 \quad \Longrightarrow \quad F (\vec X, t) = \mathbb{C} }[/math]

where [math]\displaystyle{ \mathbb{C} = \mbox{constant} }[/math]

Bernovlli's equation follows:

[math]\displaystyle{ \frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{\rho} + g Z = \mathbb{C} }[/math]

or

[math]\displaystyle{ \frac{P}{\rho} = - \frac{\partial\Phi}{\partial t}-\frac{1}{2}\nabla\Phi \cdot \nabla \Phi - g Z + \mathbb{C} }[/math]

The value of the constant [math]\displaystyle{ \mathbb{C} }[/math] is immaterial as will be shown below.

[math]\displaystyle{ \bullet }[/math] Angular momentum conservation principle contained in:

[math]\displaystyle{ \nabla \times \vec V = 0 }[/math]

-- If particles are modelled as spheres, above equation implies no angular velocity at all times.

Derivation of Nonlinear Free-surface Condition

[math]\displaystyle{ \bullet }[/math] Method I: on [math]\displaystyle{ Z=\zeta; \ P=P_a \equiv \mbox{Atmospheric Pressure} }[/math]

From Bernoulli:

[math]\displaystyle{ \longrightarrow \frac{P_a}{\rho}=-\frac{\partial\Phi}{\partial t}-\frac{1}{2}\nabla\Phi\cdot\nabla\Phi-g\zeta+\mathbb{C} \qquad \mbox{on} \ Z=\zeta(X,Y,t) }[/math]

On [math]\displaystyle{ Z=\zeta }[/math] The mathematical function

[math]\displaystyle{ Z-\zeta(X,Y,t)\equiv\tilde{f}(X,Y,Z,t) }[/math]

is always zero when tracing a fluid particle on the free surface. So the substantial or total derivative of [math]\displaystyle{ \tilde{f} }[/math] must vanish, thus

[math]\displaystyle{ \frac{D\tilde{f}}{Dt}=0=\left ( \frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) \tilde{f}=0, \qquad \mbox{on} \ Z=\zeta }[/math]

Expanding we obtain:

[math]\displaystyle{ \left (\frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) (Z-\zeta) =0, \qquad \mbox{on} \ Z=\zeta }[/math]

[math]\displaystyle{ \longrightarrow \frac{\partial\zeta}{\partial t}+\frac{\partial\Phi}{\partial X} \frac{\partial\zeta}{\partial X}+\frac{\partial\Phi}{\partial Y}\frac{\partial\zeta}{\partial Y}=\frac{\partial\Phi}{\partial Z}, \ Z=\zeta \qquad \mbox{(Kinematic free-surface condition)} }[/math]

From Bernoulli's equation we obtain the dynamic free surface condition:

[math]\displaystyle{ \longrightarrow \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla \Phi \cdot \nabla \Phi + g \zeta = \mathbb{C} - \frac{P_a}{\rho}, \ Z=\zeta \qquad \mbox{(Dynamic free-surface condition)} }[/math]