p_rgh instead of p

Tobi · May 16, 2014, 07:24

Hello all,

I have a question due to a section in the book:

Code:

Numerische Strömungsmechanik, J.H. Ferziger, M. Peric

The section is very at the beginning of the book. The momentum equation are presented and then the stress tensor is descirbed.

If only gravity is available as body force you can write the momentum eqn as follow:

$\frac{\partial (\rho u_i)}{\partial t} + \frac{\partial (\rho u_j u_i)}{\partial x_j} = \frac{\partial \tau_{ij}}{\partial x_j} - \frac{\partial p}{\partial x_i} + \rho g_i$

Then its written, that in case of constant density and gravity the last term can be written as:

$\nabla (\rho g \cdot r)$

r: position vector

Normally the gravity is supposed to act to the negativ z direction. Hence this is correct you can say that g = g_z and g_z is negativ (-9,81).

Therefor r is the z direction and then you can write:

$\nabla (\rho g_z \cdot z)$

They call it hydrostatic pressure but this is only true if the point of reference is correct (am I right?).

After that they introduced a new pressure:

$\tilde p = p - \rho g\cdot r = p - \rho g_z z$

This pressure (tilde) is called work pressure. They write that the term

$\rho g_i$
dissapear in the first equation if you use the new pressure field.

So in my case I dont understand why the last term will dissapear?

I understand that this would be nice because of numerical threatment but at the moment I dont get the point how to change the first eqn. that the last term will dissapear.

Hope its clear what I mean (:

FMDenaro · May 16, 2014, 07:38

disappears in the sense that the new pressure take into account for its contribution and you have only the gradient of the new pressure

Tobi · May 19, 2014, 08:24

Hi,

thanks for your answer, but still I dont get the point

May 16, 2014, 07:24	p_rgh instead of p	#1
Tobi Super Moderator Tobias Holzmann Join Date: Oct 2010 Location: Tussenhausen Posts: 2,708 Blog Entries: 6 Rep Power: 51	Hello all, I have a question due to a section in the book: Code: Numerische Strömungsmechanik, J.H. Ferziger, M. Peric The section is very at the beginning of the book. The momentum equation are presented and then the stress tensor is descirbed. If only gravity is available as body force you can write the momentum eqn as follow: $\frac{\partial (\rho u_i)}{\partial t} + \frac{\partial (\rho u_j u_i)}{\partial x_j} = \frac{\partial \tau_{ij}}{\partial x_j} - \frac{\partial p}{\partial x_i} + \rho g_i$ Then its written, that in case of constant density and gravity the last term can be written as: $\nabla (\rho g \cdot r)$ r: position vector Normally the gravity is supposed to act to the negativ z direction. Hence this is correct you can say that g = g_z and g_z is negativ (-9,81). Therefor r is the z direction and then you can write: $\nabla (\rho g_z \cdot z)$ They call it hydrostatic pressure but this is only true if the point of reference is correct (am I right?). After that they introduced a new pressure: $\tilde p = p - \rho g\cdot r = p - \rho g_z z$ This pressure (tilde) is called work pressure. They write that the term $\rho g_i$ dissapear in the first equation if you use the new pressure field. So in my case I dont understand why the last term will dissapear? I understand that this would be nice because of numerical threatment but at the moment I dont get the point how to change the first eqn. that the last term will dissapear. Hope its clear what I mean (: __________________ Keep foaming, Tobias Holzmann

May 19, 2014, 08:24		#3
Tobi Super Moderator Tobias Holzmann Join Date: Oct 2010 Location: Tussenhausen Posts: 2,708 Blog Entries: 6 Rep Power: 51	Hi, thanks for your answer, but still I dont get the point __________________ Keep foaming, Tobias Holzmann

May 16, 2014, 07:38		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,773 Rep Power: 71	disappears in the sense that the new pressure take into account for its contribution and you have only the gradient of the new pressure