Volume Flux for a Hydraulic Jump

In summary, the conversation involves a problem that is given in an image and the use of the continuity or conservation of mass equation to solve it. The conversation includes attempts at integrating the equation with respect to x and z, using the Leibniz rule, and discussing the possibility of h1 and h2 being equal, which does not make sense in the context of a hydraulic jump. The speaker is unsure about their math and asks for help in finding where they may have gone wrong.
  • #1
Particle Head

Homework Statement


Problem is given in this image,
https://gyazo.com/454370ff9549dcd7c53604ebfe5df105

Homework Equations



Continuity or conservation of mass equation:
[tex] \frac{\partial u}{\partial x} + \frac{\partial w}{\partial z} = 0 [/tex]
Where u is the horizontal velocity and w is the vertical velocity

The Attempt at a Solution



Firstly I integrated the conservation of mass equation with respect to x between the two points:

[tex] \int_ {x_1} ^ {x_2} \frac{\partial u}{\partial x} \mathrm{d}x + \int_ {x_1} ^ {x_2} \frac{\partial w}{\partial z} \mathrm{d}x = 0
[/tex]
Which after evaluating I get,
[tex] u(x_2) - u(x_1) + x_2 \frac{\partial w}{\partial z} - x_1 \frac{\partial w}{\partial z} = 0 [/tex]
Firstly here I am not 100% sure I can assume w is just a function of z only, but I have yet to see it as a function of anything else other than t in 2D flow?

I then integrated the mass equation vertically first from [itex] z = 0 [/itex] to [itex] z = h_1 [/itex] and then from [itex] z=0 [/itex] to [itex] z= h_2 [/itex]

[tex] \int_{0}^{h_1} \frac{\partial u}{\partial x} dz + \int_{0}^{h_1} \frac{\partial w}{\partial z} dz = 0 [/tex]

which yields,
[tex] h_1 \frac{\partial u}{\partial x} - w(0) = 0 [/tex]
Since [tex] w(h_1) = 0 [/tex]

Similarly for [itex] z=0 [/itex] to [itex] z= h_2 [/itex] I get,
[tex] h_2 \frac{\partial u}{\partial x} - w(0) = 0 [/tex]

This is where I am kind of stuck, from these two equations it appears [itex] h_1 = h_2 [/itex] which doesn't make sense since this is a hydraulic jump and the nature of such is to increase the surface of the liquid.

Any hints where I may be going wrong here or missing something is appreciated.

Thanks.
 
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  • #2
First integrate with respect to z, using the Leibnitz rule for differentiation under the integral sign on the u term.
 
  • #3
I try to integrate with respect to z first, I haven't used the Leibniz rule before but from what I can gather I can treat my bounds as constant thus I get this for the u term.

[tex] \int_{0}^{h_1} \frac {\partial u}{\partial x} dz = \frac {\partial}{\partial x} \int_{0}^{h_1} u dz [/tex]

After evaluating this I get,
[tex] h_1 \frac {\partial u}{\partial x} [/tex]

I'm not sure if I did this correct but following this doesn't seem to get me much further?
I suspect my math is off I haven't done any multi variable calc for a while.
 
  • #4
$$\frac{\partial [\int_0^{h(x)}udz]}{\partial x}=\int_0^{h(x)}{\frac{\partial u}{\partial x}dz}+u(x,h)\frac{dh}{dx}$$
 

1. What is volume flux for a hydraulic jump?

Volume flux is a measure of the amount of water that is flowing through a hydraulic jump. It is typically expressed in cubic meters per second (m³/s) and is an important parameter in understanding the behavior of water in a hydraulic jump.

2. How is volume flux calculated for a hydraulic jump?

Volume flux can be calculated by multiplying the cross-sectional area of the water at the jump by the velocity of the water. This gives the volume of water passing through a specific point in the jump per unit time.

3. What factors affect volume flux in a hydraulic jump?

The main factors that affect volume flux in a hydraulic jump are the upstream flow rate, the geometry of the jump, and the roughness of the channel bed. Changes in any of these factors can impact the volume flux.

4. Why is volume flux important in hydraulic jump analysis?

Volume flux is important in hydraulic jump analysis because it helps determine the energy dissipation and flow characteristics of the jump. It also helps in designing and managing hydraulic structures, such as spillways and weirs, to ensure safety and efficiency.

5. How can volume flux be controlled in a hydraulic jump?

Volume flux can be controlled in a hydraulic jump by adjusting the geometry of the jump, such as the depth and width of the channel, to achieve the desired flow rate. Additionally, using flow control structures, such as baffles and energy dissipaters, can also help regulate the volume flux in a hydraulic jump.

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