Workshop 01

Warning

These functions were provided for the in-class session on Friday, April 26 and are used in the solution (next page). See the note below about changes made to the third function after the in-class session.

import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as st

def wind_set_up(u, d, F=3000, C=4.0e-6):
    """
    Calculate wind set-up at the toe of the dike, relative to still water level.

    Parameters
    ----------
    u : float
        [m/s] Representative wind speed
    d : float
        [m] Average depth over fetch length. Take "offshore" depth as proxy.
    F : float
        [m] Fetch length, default is 3000 m (3 km)
    C : float, optional
        [-] Empirical constant, between 3.5e-6 and 4.0e-6, default is 4.0e-6
    
    Returns
    -------
    S : float
        [m] Wind set-up at toe of the dike
    """

    g = 9.81
    S = C*(u**2)/(g*d)*F  
    return S

def run_up_level(Hs, alpha):
    import numpy as np
    """
    Calculate run-up level at the toe of the dike, relative to still water level.

    Parameters
    ----------
    Hs : float
        [m] Significant wave height
    alpha : float
        [degrees] Angle of the slope of the toe of dike
    
    Returns
    -------
    Ru2 : float
        [m] Run-up level, exceeded by 2% of the waves
    """
    # Since numpy works with radians, we need to convert the angle to radians
    alpha_radians = np.radians(alpha)

    Ru2 = 8*Hs*np.tan(alpha_radians)
    return Ru2

Warning

The function below was modified after the in-class session on April 26; the name is changed to make this more apparent. There are two primary changes:

1. The keyword argument d_average was replaced with both elevations as random variables (Zw and Zb). This allows the “offshore” water elevation and water depth to be used separately in the “complete” equation, rather than the water depth only (using this in place of Zw underestimates the water elevation at the toe of the dike, since \(Z_w>d_{w,\,\mathrm{dike}}\)).

2. The keyword argument d_average was set to a value of 3, so if this was used, you were effectively ignoring the variability of the tide and bottom elevation. This had a small effect on the calculated failure probability: implementing the new function caused it to change from 8% to 37%.

def water_at_dike(Hs, u, Zw, Zb, alpha=20, F=10000, C2 = 4.0e-6):
    """
    Calculate the effective water elevation at the dike.

    Parameters
    ----------
    Hs : float
        [m] Significant wave height
    alpha : float
        [degrees] Angle of the slope of the toe of dike
    u : float
        [m/s] Representative wind speed
    d : float
        [m] Average depth over fetch length
    F : float
        [m] Fetch length
    Zw : float
        [m] Offshore water elevation (tidal level)
    Zb : float
        [m] Offshore bed level (or bottom level)
    
    Intermediate Parameter
    ----------------------
    d_offshore : float
        [m] Depth of water at the offshore location, Zw - Zb

    Returns
    -------
    Zd : float
        [m] Elevation of water level at the dike.
              - Equivalent to water depth, as datum is toe of dike
              - Sum of tidal level, run-up, wind set-up
    """

    d_offshore = Zw - Zb

    S = wind_set_up(u, d_offshore, F, C2)

    Ru2 = run_up_level(Hs, alpha)
    
    Zd = Zw + Ru2 + S

    return Zd