Exercise 8

Shape factor

The shallow water equations with velocity profile correction in one space dimension are given by:

\[ \frac{\partial}{\partial t} \begin{pmatrix} h \\ h u \end{pmatrix} + \frac{\partial}{\partial x} \begin{pmatrix} h u \\ \alpha h u^2 + g \frac{h^2}{2} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \]

where \(h(x,t)\) describes the fluid depth, \(u(x,t)\) the mean flow velocity and \(g=9.81 \textrm{m}/\textrm{s}^2\) is the gravitational acceleration.

  1. What does \(\alpha\) stand for and how can it be computed? Determine \(\alpha\) for a linear velocity profile with \(u(y=0) = 0\) and \(u(y=h) = u_s\).

  2. Assume a shallow lake, currently at rest. That means \(h=H\) and \(u=0\). A stone is thrown into the lake and causes a small surface perturbation in height and velocity, denoted by \(h_1\) and \(u_1\). Derive a of equations for \(h_1\) and \(u_1\) by means of linearization.

Tip

The linearization can be performed by looking at small perturbations from a ground state \((h_0, u_0)\), e.g. by defining: \(h(x,t) = h_0(x, t) + \delta \, h_1(x,t) + \mathcal{O}( \delta^2 )\) and \(u(x,t) = u_0(x, t) + \delta \, u_1(x,t) + \mathcal{O}( \delta^2 )\), where \(\delta\) denotes the small amplitude of the perturbation.

  1. Reduce the system derived in Task 2 to an equation for \(h_1\). What type of equation do we find?

  2. The wave ansatz is given by \(h_1(x,t) = A \, e^{i (k \, x - \omega \, t)}\). State what \(A\), \(i\), \(k\) and \(\omega\) stand for? Using the wave ansatz, compute the phase velocity.

  3. Compute the eigenvalues of the shallow water equation for \(\alpha=1\). Explain the differences and similarities between the eigenvalues and the phase velocity.

  4. Assume that we wait \(10s\) until a surface perturbation caused by a stone reaches the lake’s shore. How much longer would we have to wait at a similarly sized lake on Saturn’s moon Titan? (Titan has surface lakes of liquid ethane and methane and a gravitational acceleration of \(g=1.4 \frac{m}{s^2}\))

Scaling of the shallow water equations for complex materials

The following equation describe the incompressible Navier-Stokes equations in dimensionless form using a scaling with dimensionless numbers in \((\epsilon, Fr)\). The system uses a rotated coordinate system with inclination angle \(\zeta\):

\[ \begin{aligned} & \partial_x u + \partial_z w = 0 \\ & \epsilon Fr^2 \left( \partial_t u + \partial_x u^2 + \partial_z (uw) \right) = \epsilon \partial_x \sigma_{xx} + \partial_z \sigma_{xz} + \sin(\zeta) \\ & \epsilon^2 Fr^2 \left( \partial_t w + \partial_x (uw) + \partial_z (w^2) \right) = \epsilon \partial_x \sigma_{xz} + \partial_z \sigma_{zz} - \cos(\zeta) \; . \end{aligned} \tag{1}\]

We use a material model of the form:

\[ \begin{aligned} \boldsymbol{\sigma} = \begin{pmatrix} \sigma_{xx} & \sigma_{xz} \\ \sigma_{zx} & \sigma_{zz} \end{pmatrix} = - p \boldsymbol{I} + \begin{pmatrix} 0 & \tau_{xz} \\ \tau_{xz} & 0 \end{pmatrix} \; . \end{aligned} \tag{2}\]

In the following, we denote \(b(x)\) as the bottom topography, \(s(t,x)\) as the free surface and \(h(t,x)=s(t,x)-b(x)\) as the water height.

Assume the shear stress \(\tau_{xz}\) is given by

\[ \tau_{xz} = - \epsilon^{-1} \nu \partial_z u - \epsilon^0 \sin(\zeta) (z-b) \tag{3}\]

with a linear velocity profile

\[ u(t,x,z) = \hat{u}(t,x) \frac{z-b(x)}{h(t,x)} \tag{4}\]

where \(\hat{u}(t,x) = u(t,x,z=s(t,x))\) denotes the velocity at the free-surface.

  1. Derive an expression for the shear stress \(\tau_{xz}\) (Equation 3) using the velocity profile (Equation 4).

  1. Substitute your result from task 1. and the material model (Equation 2) into Equation 1. and simplify the resulting system.

  2. Now we consider the ansatz

\[ \begin{aligned} & u = \epsilon^0 u^{(0)} + \epsilon^1 u^{(1)} + \mathcal{O}(\epsilon^2) \\ & w = \epsilon^0 w^{(0)} + \epsilon^1 w^{(1)} + \mathcal{O}(\epsilon^2) \\ & p = \epsilon^0 p^{(0)} + \epsilon^1 p^{(1)} + \mathcal{O}(\epsilon^2). \end{aligned} \]

Substitute it into the expression of your simplified system (task 2.) and neglect terms of order \(\epsilon^2\) or higher.

  1. Write down the asymptotic expansion by separating the system according to its scales. Your result should yield non-trivial equations on the scales \(\left(\epsilon^{0}\right)\) and \(\left(\epsilon^{1}\right)\).

The pressure distribution now takes the form

\[ p(t,x,z) = A(t,x) (h-(z-b)) \tag{5}\]

  1. Compute \(A(t,x)\) by first deriving equations for \(p^{(0)}\) and \(p^{(1)}\) and assuming a stress-free free-surface boundary condition \(p(t,x,z=s) = 0\).

  2. Is the pressure hydrostatic? Explain your answer.