Shallow water equations evaluation

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In summary, there is confusion regarding the momentum equation and the conservative shallow water equations. While using product rules, the first line can be easily obtained, but the second line is causing difficulties. The issue is further illustrated in a paper that presents the same problem. The paper states that the second set of equations is non-conservative, but it is unclear how this was derived from the original equations. The Wikipedia entry on shallow water equations is also not helpful in explaining this. It is unclear how expanding the terms can result in a non-conservative set of equations, as the underlying assumptions used to derive them remain the same.
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maistral
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Summary:: A little confusion on the momentum equation (I think).

According to Wikipedia (I know, I just need basic resources for now), the conservative SWEs are
:
9b9d481407c0c835525291740de8d1c446265ce2


If I use product rules, I am supposed to get:

6bb10fdfb320a6bc0f4011b08b6b616b2a95929e


For context, note that ρ is a constant and can be taken out (thus canceled out), and η(x,y) = H + h(x,y) (H is a constant).

I have no issues getting the first line. The second line however, I am facing issues. This is what I have:

1638011369713.png


I intentionally did not distribute the derivatives with respect to y so I could see the problem more clearer. Am I missing something here with regards to the expansion of the PDE? Thanks in advance.
 
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  • #2
Just an update:

I found this paper that actually is the same 'issue' that I am facing, though they just presented it.

This paper:
https://doi.org/10.1063/5.0039545

It stated that:

1638033387994.png


I am having trouble understanding how the second equations have been made (second equation 4, to second equation 5).
 
  • #3
The wikipedia entry on shallow water equations doesn't make sense to me. How could you, from equations derived using momentum and mass conservation, merely expand terms and get a non-conservative set of equations. No mechanical manipulation of equations changes the underlying assumptions they were derived from.
 

1. What are the shallow water equations?

The shallow water equations are a set of partial differential equations that describe the behavior of shallow water waves, such as those found in oceans, lakes, and rivers. They take into account the effects of gravity, pressure, and friction on the water's surface.

2. Why are the shallow water equations important?

The shallow water equations are important because they allow scientists to accurately model and predict the behavior of water in shallow areas. This is crucial for understanding and predicting the effects of natural disasters such as tsunamis, storm surges, and floods.

3. How are the shallow water equations evaluated?

The shallow water equations are typically evaluated using numerical methods, such as finite difference or finite volume methods. These methods involve breaking down the equations into smaller, discrete parts and solving them iteratively to approximate the behavior of the water over time.

4. What are some applications of the shallow water equations?

The shallow water equations have a wide range of applications, including oceanography, meteorology, coastal engineering, and hydrology. They are used to study and predict the behavior of water in coastal areas, rivers, and lakes, and to develop strategies for mitigating the effects of natural disasters.

5. What are the limitations of the shallow water equations?

While the shallow water equations are useful for modeling and predicting the behavior of water in shallow areas, they have some limitations. They do not take into account the effects of turbulence or small-scale features, and they assume that the water is incompressible and inviscid. Additionally, they may not accurately represent the behavior of water in more complex environments.

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