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I might play around with the linear energy balance just mentioned and see if I could get them to be within a factor of two or three and decide I was doing something wrong in parameterizing if I couldn’t, but then I’d permanently abandon calculating myself and turn to this, ir something similar:

I might play around with the linear energy balance just mentioned and see if I could get them to be within a factor of two or three and decide I was doing something wrong in parameterizing if I couldn’t, but then I’d permanently abandon calculating myself and turn to this, or something similar:

This is hard

The parameters defining the fluid are viscosity and density. A secondary or even tertiary factor is resistance to rotation. Fluid analysis sometimes assumes inviscid, irrotational, or both.

Here, in an attempt to mitigate the fact that water doesn’t scale, you would use liquids with different densities and viscosities depending upon the size of the scale. You could build a one-eighth scale model and use water that has been “scaled” by a factor of eight. I don’t know what liquid it is, but there is one that is one eighth water and that is a common scale built. I searched a but and couldn’t find anything, but for what it’s worth I do remember there is a “one eighth water” out there that people use.

The total energy per wave per unit width of crest, E, using same variables as above, is: $$E = \tfrac{1}{8} \rho g H^2 L$$

Even here, it is already nonlinear (quadratic) in height.

Where $Q$ is flowrate (like gpm) and $A$ is the cross-sectional area of the stream where it hits the surface. (Unfortunately that would change, even in a relative way, with scaling too). Presumably doubling this would increase $Q$ a factor of $2 \sqrt{2}$ and $A$ fourfold.

The next form is the Boussinesq approximation. Stokes wave equation will be soon needed to determine how the waves scale because we can’t just keep the same height/length ratio and increase its size; at first we could assume wavelength and height scale similarly, so doubling those would increase energy by a factor of $8$, because $(2H)^2 ~ (2L)~ =~8H^2 ~ L$. But this is damped with increasing depth.

You can see why computational methods are employed for even basic problems. It’s basically not worth doing anything analytically on paper. The reason students do such problems is so they are not stupid about using cfd, for estimating, deciding what matters which might affect how much effort is put into measuring things in the field, for not accepting obviously ridiculous results uncritically, etc. Here is a video of someone using ANSYS for waves in an open channel:

This is difficult

The most important parameters defining the fluid are viscosity and density. A secondary or even tertiary factor is resistance to rotation. Fluid analysis sometimes assumes inviscid, irrotational, or both.

Next, in an attempt to mitigate the fact that water doesn’t scale, you would use liquids with different densities and viscosities depending upon the size of the scale. You could build a one-eighth scale model and use water that has been “scaled” by a factor of eight. I don’t know what liquid it is, but there is one that is one eighth water and that is a common scale built. I searched a bit and couldn’t find anything, but for what it’s worth I do remember there is a “one eighth water” out there that people use.

The total energy per wave per unit width, E, using same variables as above, is: $$E_{out} = \tfrac{1}{8} \rho g H^2 l$$

Multiply by the arc length of the coherent wave about three waves into it (where you have the good, clear $2cm$ wave) because the above is energy per wave per unit width, and by frequency (waves per second to get power, which will be equal to power input, next equation).

Where $Q$ is flowrate (like gpm) and $A$ is the cross-sectional area of the stream where it hits the surface. (Unfortunately that would change, even in a relative way, with scaling too). Presumably doubling this would increase $Q$ a factor of $2 \sqrt{2}$ and $A$ fourfold. Some of the energy of the stream will dissipate below the surface. This is on the order of half for a wide range, but that doesn’t matter because you will need to make the energies agree. (i.e. you have actual values for everything above, so the two energies calculated can be made to agree by adjusting the $0.5$ just mentioned and then assuming that constant as you scale).

The next more accurate form, for including second-order wave effects, is the Boussinesq approximation. Stokes wave equation will be soon needed to determine how the waves scale because we can’t just keep the same height/length ratio and increase its size; at first we could assume wavelength and height scale similarly, so doubling those would increase energy by a factor of $8$, because $(2H)^2 ~ (2l)~ =~8H^2 ~ l$. Finally we assumed it was deep enough that more depth will not provide further damping, meaning we assumed the constant ($0.5$ or whatever it was calculated to be) doesn’t change going to the mew scale.

You can see why computational methods are employed for even basic problems. It’s basically not worth doing anything analytically on paper. The reason students do such problems at all is so they are not stupid about using cfd, for estimating, for deciding what matters which might affect how much effort is put into measuring things in the field, for not accepting obviously ridiculous results uncritically, etc. Here is a video of someone using ANSYS for waves in an open channel:

I might play around with the linear energy balance just mentioned and see if I could get them to be within a factor of two or three and decide I was doing something wrong in parameterizing if I couldn’t, but then I’d permanently abandon calculating myself and turn to this, ir something similar:

The next form is the Boussinesq approximation. Stokes wave equation will be soon needed to determine how the waves scale; at first we could assume wavelength and height scale similarly, so doubling those would increase energy by a factor of $8$, because $(2H)^2 ~ (2L)~ =~8H^2 ~ L$. But this is damped with increasing depth.

The next form is the Boussinesq approximation. Stokes wave equation will be soon needed to determine how the waves scale because we can’t just keep the same height/length ratio and increase its size; at first we could assume wavelength and height scale similarly, so doubling those would increase energy by a factor of $8$, because $(2H)^2 ~ (2L)~ =~8H^2 ~ L$. But this is damped with increasing depth.