$0\mathrm{.}21-$D Shallow$-$Water Model

Figure $1$: Configuration of the one$-$dimensional shallow$-$water model. Undisturbed water depth is

h$0.$

Governing equations for a $1-$D shallow$-$water model $($see Figure $1)$ can be written as:

$$u

$$t $=$g $\backslash$eta

$$x $(1)$

$\backslash$eta

$$t $=($u h$)$

$$x $(2)$

where u is speed in the x$-$direction, t is time, g is acceleration due to gravity, $\backslash$eta is sea$-$level elevation,

and h is the total water depth.

Analytical Solution

The water depth h can be approximated as constant for a flat seafloor together with wave amplitudes

small compared with total water depth. In this case, the wave solution of the above equation

is:

$\backslash$eta $($t$,$ x$)=\backslash$eta $0$ sin$(2\backslash$pi x$/\backslash$lambda $2\backslash$pi t$/$T $)(3)$

u$($t$,$ x$)=$ u$0$ sin$(2\backslash$pi x$/\backslash$lambda $2\backslash$pi t$/$T $)(4)$

Page $1$ of $4$

Final Take$-$Home Portion, Numerical Methods $($CWR$3561),$ Spring $2024$

Instructor: Dr$.$ C$.$ i $$gdem Akan

Due: $1$ May by $11$:$59$ pm

where $\backslash$eta $0$ is wave amplitude, $\backslash$lambda is wavelength, T is wave period, and the magnitude of u is given

by:

u$0=\backslash$eta $0$

pg$/$h $(5)$

Horizontal flow under a long surface wave is depth$-$independent and so are horizontal gradients of

u$.$ The solution for vertical speed of a fluid parcel as a function of depth can be written as:

w$($t$,$ x$,$ z$)=2\backslash$pi u$0$ z$/\backslash$lambda cos$(2\backslash$pi x$/\backslash$lambda $2\backslash$pi t$/$T $)(6)$

where z$$ is $($positive$)$ distance from seafloor.

$0\mathrm{.}3$ MATLAB Coding

You will create an animation of analytical wave solution using the input values below:

$$ Wave amplitude $(\backslash$eta $0)=1$ m

$$ Wavelength $(\backslash$lambda $)=500$ m

$$ Water depth $($h$)=20$ m

$$ Wave speed $($c$)=$gh

$$ Wave period $($T $)=\backslash$lambda $/$c

In order to generate the animation, you will need to track individual fluid parcels using the equations

below:

xpnew $=$ xpold $+$ dt $\backslash$times u $(7)$

zpnew $=$ zpold $+$ dt $\backslash$times w $(8)$

You will track the particles at zp $=1\mathrm{.}0$ m$,$ zp $=6\mathrm{.}0$ m$,$ zp $=11\mathrm{.}0$ m and zp $=16\mathrm{.}0$ m

Suggested Algorithm

$$ Enter the input variables

$$ Limits of x$-$axis: $[0,2\backslash$lambda $].$ Use $21$ points.

$$ Number of time steps $=100.$ Total animation time $=2\backslash$times T

$$ Calculate dt

$$ Initialize particle positions

$$ You only need one loop: loop over number of time steps.

$$ Calculate $\backslash$eta

$$ Calculate u

$$ Calculate w

$$ Calculate the new fluid parcel locations

Page $2$ of $4$

Final Take$-$Home Portion, Numerical Methods $($CWR$3561),$ Spring $2024$

Instructor: Dr$.$ C$.$ i $$gdem Akan

Due: $1$ May by $11$:$59$ pm

$$ Plot all the particles

$*$ Show the water surface with a thick solid line

$*$ Show rest of the particles with a circle

$*$ Label the x$-$ and y$-$axis properly

$*$ Use sprintf command to show the time at each time step

$$ Save the frame

$$ Create the animated GIF

Animation with MATLAB

There are several ways to create animated gif files with MATLAB. For example, examine the code

provided in class. You can adapt it and use it for your code.

$0\mathrm{.}4$ Hand Calculations

Calculate the following for x $=50$ m$,$ z $=16$ m and x $=300$ m$,$ z $=6$ m at t $=10$ s:

$$ Wave speed $($c$)=$gh

$$ Wave period $($T $)=\backslash$lambda $/$c

$\backslash$eta

$$ u

$$ w

Midterm Take Home Rubric

$$ Title Page: This should consist of a blank page with your name, UNF ID$,$ course title, home$-$

work assignment number and title. $5$ pts

$$ Summary$/$Observations: This should be a brief summary of the assignment. What did the

assignment ask for? What did you learn? Did you discover anything easy$/$difficult with the

assignment? It doesn$$t have to be long, just clear and concise. Do not write only one

sentence. $10$ pts

$$ MATLAB Code$($s$)$: Copy and paste your code$($s$)$ into the Word document. $40$ pts

$$ Output: Animation file in $.$gif format $20$ pts

$$ Hand Calculations: See the hand calculations section. $25$ pts

Extra Credit

Recall the $$Ready$,$ Aim, Fire$$ GUI we used in class. Make improvements to the GUI interface and

the code by applying one of the options below:

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