Layer of fluid between two pliant surface

Luc_27 · November 21, 2018, 11:12

Hello everybody.

I am quite new to the fluid dynamics world and I am currently facing a problem, which I do not understand.
By following the passages done on some papers and books, I have to derive the general depth-averaged flow equations in a layer of fluid. I am using the incompressible, three-dimensional Reynolds-Averaged Navier-Stokes equations for the atmospheric boundary layer (which I do not report here since not directly related to what I do not understand).

The notation used is as follow:
- $\overline{\textbf{u}}=(\overline{u},\overline{v},\overline{w})$
- the bar denotes that the quantity is averaged over time

The first passage I do not understand is the following:
''Consider now a layer of fluid between two pliant surfaces

and

that follow a streamline, for which:
$\begin{aligned} \overline{w}(z_a) &= \partial_t z_a + \overline{u}(z_a)\partial_x z_a +\overline{v}(z_a)\partial_y z_a \\ \overline{w}(z_b) &= \partial_t z_b + \overline{u}(z_b)\partial_x z_b +\overline{v}(z_b)\partial_y z_b \end{aligned}$
What does it mean? Which is the path that brings to these formula? Which physic interpretation can be done?

The second passage I do not understand is the following:
''Since

and

lie on a streamline, then:
$\begin{aligned} \overline{u}(z_b) \overline{u}(z_b) \partial_x z_b &= \overline{u}(z_a) \overline{u}(z_a) \partial_x z_a \\ \overline{u}(z_b) \overline{v}(z_b) \partial_y z_b &= \overline{u}(z_a) \overline{v}(z_a) \partial_y z_a \end{aligned}$
I simply do not understand why these quantities are equal.

Thank you in advance for the help.

FMDenaro · November 21, 2018, 13:41

Please, link the original textbook you are using

Luc_27 · November 22, 2018, 03:09

Unluckily it is written on a paper that is still not published, so I do not think that I can post it here. But anyway, what I do not understand are exactly that two passages, and in the papers they are reported with these words, no more explanation.

I did search on books or other materials, but I do not find nothing similar. Like I said, I am speaking about a layer of fluid between two pliant surface,

and

, and then that equations just pop up.

Whatever kind of help is appreciated, thank you.

Luc_27 · November 25, 2018, 06:04

Hi guys,

I did solve all problems, it was all about mathematics and some physic assumption.

By the way, I have another question. What does it really mean depth-averaged flow equations? Because I am working with a model that deals with atmospheric boundary layer; this is split in three levels, and in each level there are different 3-D Navier-Stokes equations. Then it is made a depth averaged over the depth of each level, and the system becomes 2-D.

Does this means that in each level all quantities along the z-direction are constant since we have averaged it?

Probably it is like this, but to me it seems a very strong assumption.

FMDenaro · November 25, 2018, 06:31

Quote:

Originally Posted by Luc_27

Hi guys,

I did solve all problems, it was all about mathematics and some physic assumption.

By the way, I have another question. What does it really mean depth-averaged flow equations? Because I am working with a model that deals with atmospheric boundary layer; this is split in three levels, and in each level there are different 3-D Navier-Stokes equations. Then it is made a depth averaged over the depth of each level, and the system becomes 2-D.

Does this means that in each level all quantities along the z-direction are constant since we have averaged it?

Probably it is like this, but to me it seems a very strong assumption.

I think that the hypothesis is that the contribution along the depth is disregardable.

Maybe you can find the answer here
https://coast.nd.edu/jjwteach/www/ww...s/topic1_5.pdf

November 21, 2018, 11:12	Layer of fluid between two pliant surface	#1
Luc_27 New Member Join Date: Nov 2018 Posts: 10 Rep Power: 7	Hello everybody. I am quite new to the fluid dynamics world and I am currently facing a problem, which I do not understand. By following the passages done on some papers and books, I have to derive the general depth-averaged flow equations in a layer of fluid. I am using the incompressible, three-dimensional Reynolds-Averaged Navier-Stokes equations for the atmospheric boundary layer (which I do not report here since not directly related to what I do not understand). The notation used is as follow: - $\overline{\textbf{u}}=(\overline{u},\overline{v},\overline{w})$ - the bar denotes that the quantity is averaged over time The first passage I do not understand is the following: ''Consider now a layer of fluid between two pliant surfaces and that follow a streamline, for which: $\begin{aligned} \overline{w}(z_a) &= \partial_t z_a + \overline{u}(z_a)\partial_x z_a +\overline{v}(z_a)\partial_y z_a \\ \overline{w}(z_b) &= \partial_t z_b + \overline{u}(z_b)\partial_x z_b +\overline{v}(z_b)\partial_y z_b \end{aligned}$ What does it mean? Which is the path that brings to these formula? Which physic interpretation can be done? The second passage I do not understand is the following: ''Since and lie on a streamline, then: $\begin{aligned} \overline{u}(z_b) \overline{u}(z_b) \partial_x z_b &= \overline{u}(z_a) \overline{u}(z_a) \partial_x z_a \\ \overline{u}(z_b) \overline{v}(z_b) \partial_y z_b &= \overline{u}(z_a) \overline{v}(z_a) \partial_y z_a \end{aligned}$ I simply do not understand why these quantities are equal. Thank you in advance for the help.

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November 21, 2018, 13:41		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,783 Rep Power: 71	Please, link the original textbook you are using

November 22, 2018, 03:09		#3
Luc_27 New Member Join Date: Nov 2018 Posts: 10 Rep Power: 7	Unluckily it is written on a paper that is still not published, so I do not think that I can post it here. But anyway, what I do not understand are exactly that two passages, and in the papers they are reported with these words, no more explanation. I did search on books or other materials, but I do not find nothing similar. Like I said, I am speaking about a layer of fluid between two pliant surface, and , and then that equations just pop up. Whatever kind of help is appreciated, thank you.

November 25, 2018, 06:04		#4
Luc_27 New Member Join Date: Nov 2018 Posts: 10 Rep Power: 7	Hi guys, I did solve all problems, it was all about mathematics and some physic assumption. By the way, I have another question. What does it really mean depth-averaged flow equations? Because I am working with a model that deals with atmospheric boundary layer; this is split in three levels, and in each level there are different 3-D Navier-Stokes equations. Then it is made a depth averaged over the depth of each level, and the system becomes 2-D. Does this means that in each level all quantities along the z-direction are constant since we have averaged it? Probably it is like this, but to me it seems a very strong assumption.