Discretization of mass conservation equation in a 2D staggered grid

kiwiguigou · April 15, 2016, 15:34

Hi everybody,

As mentioned in the title I would like to solve the mass conservation to simulate the boiling phenomenon into a tube bundle. I am facing some issues to discretize these equations into a staggered grid after to have applied an integration into a finite volume.

Accordingly to the classical staggered grid: the flow properties are content in the control volume whereas the velocity information is on the bondaries of this control volume. Here is a representation of the staggered grid I use (for the notations)

My governing equation is:

$\frac{\delta \left({\propto }_k{\rho }_k\right)}{\delta t}+\frac{\delta \left({\propto }_k{\rho }_ku_k\right)}{\delta x}+\frac{\delta \left({\propto }_k{\rho }_ku_k\right)}{\delta y}={\left(-1\right)}^k\left(\mathrm{\Gamma }e-\mathrm{\Gamma }c\right)+{\dot{M}}_{k\ in}$

Where the subcript k is for the phase type (1 for liquid; 2 for gas)

$\mathrm{\Gamma }c\$ and $\mathrm{\Gamma }e\$ are the evaporation and condensation rate defined as:

$\mathrm{\Gamma }e=\frac{{\propto }_1{\rho }_1}{{\tau }_e}.\frac{\left(h_1-h^{'}\right)}{\left(h^{''}-h^{'}\right)}$
$\mathrm{\Gamma }c=\frac{{\propto }_1{\rho }_1}{{\tau }_c}.\frac{\left(h^{'}-h_1\right)}{\left(h^{''}-h^{'}\right)}$

Where H" is the staturated enthalpy of the gas, H' is the saturated enthalpy of the liquid and $\tau$ is the relaxation time. First question: does someone have a definition to calculate that relaxation time?

After having integrated the mass conservation equation for the control volume I;J, I get:

$\left[{\left({\propto }_1{\rho }_1\right)}^n_{I;J}-{\left({\propto }_1{\rho }_1\right)}^{n-1}_{I;J}\right]\frac{\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}}{\mathrm{\Delta }\mathrm{t}}+\left[{\left({\propto }_1{\rho }_1u_1\right)}^n_{i+1;j}-{\left({\propto }_1{\rho }_1u_1\right)}^n_{i;j}\right]$ $\mathrm{\Delta }\mathrm{y}+\left[{\left({\propto }_1{\rho }_1v_1\right)}^n_{i;j+1}-{\left({\propto }_1{\rho }_1v_1\right)}^n_{i;j}\right]\mathrm{\Delta }\mathrm{x}=$ $\ {\left[\mathrm{-}\mathrm{\Gamma }e+\mathrm{\Gamma }c+{\dot{M}}_{1\ in}\right]}^n_{I;J}\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}$

By making the assumption that the velocity field is known, I can resolve this equation and find the void fraction and the density. My issue arrives now:
I don't know how to approximate the terms of void fraction and density which are at the bondaries of the control volume because the information is not stocked there (ex: ${\left({\propto }_1{\rho }_1\right)}^n_{i+1;j}$ )

If someone can help me on this I will be very grateful.

In advance, thank you.

Kiwi

FMDenaro · April 15, 2016, 17:35

your grid arrangement is not totally clear to me from the picture...

I immagine that rho(i,j), u(i,j), v(i,j) are staggered each other, right?
For example, if rho(i,j) is at the position x(i),y(j), then u(i,j) is at x(i)+dx/2,y(j) and v(i,j) is at x(i),y(j)+dy/2?

If it is true, the integration for rho is for the CV centred at x(i),y(j).

kiwiguigou · April 16, 2016, 07:26

Hi FMdenaro and thank you for your answer.

Yes you are right; if we use the notation (i,j) (i+1/2) and (j+1/2) the equation can consequenlty be written as:

$\left[{\left({\propto }_1{\rho }_1\right)}^n_{i;j}-{\left({\propto }_1{\rho }_1\right)}^{n-1}_{i;j}\right]\frac{\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}}{\mathrm{\Delta }\mathrm{t}}+\left[{\left({\propto }_1{\rho }_1u_1\right)}^n_{i+1/2;j}-{\left({\propto }_1{\rho }_1u_1\right)}^n_{i-1/2;j}\right]$ $\mathrm{\Delta }\mathrm{y}+\left[{\left({\propto }_1{\rho }_1v_1\right)}^n_{i;j+1/2}-{\left({\propto }_1{\rho }_1v_1\right)}^n_{i;j-1/2}\right]\mathrm{\Delta }\mathrm{x}=$ $\ {\left[\mathrm{-}\mathrm{\Gamma }e+\mathrm{\Gamma }c+{\dot{M}}_{1\ in}\right]}^n_{i;j}\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}$

Is it more clear for you with that writing?
So you mean that for the terms ${\left({\propto }_1{\rho }_1\right)}^n_{i+1/2;j}$ I can make the approximation $\frac{{\left({\propto }_1{\rho }_1u_1\right)}^n_{i+1;j}-{\left({\propto }_1{\rho }_1u_1\right)}^n_{i;j}}{\mathrm{\Delta }\mathrm{x}}$ ?

So the equation would become:

$\left[{\left({\propto }_1{\rho }_1\right)}^n_{i;j}-{\left({\propto }_1{\rho }_1\right)}^{n-1}_{i;j}\right]\frac{\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}}{\mathrm{\Delta }\mathrm{t}}+$ $\left[{\frac{{\left({\propto }_1{\rho }_1\right)}^n_{i+1;j}-{\left({\propto }_1{\rho }_1\right)}^n_{i;j}}{\mathrm{\Delta }\mathrm{x}}\left(u_1\right)}^n_{i+{1}/{2};j}-{\frac{{\left({\propto }_1{\rho }_1\right)}^n_{i;j}-{\left({\propto }_1{\rho }_1\right)}^n_{i-1;j}}{\mathrm{\Delta }\mathrm{x}}\left(u_1\right)}^n_{i-{1}/{2};j}\right]\mathrm{\Delta }\mathrm{y}+$ $\left[{\frac{{\left({\propto }_1{\rho }_1\right)}^n_{i;j+1}-{\left({\propto }_1{\rho }_1\right)}^n_{i;j}}{\mathrm{\Delta }\mathrm{y}}\left(v_1\right)}^n_{i;j+{1}/{2}}-{\frac{{\left({\propto }_1{\rho }_1\right)}^n_{i;j}-{\left({\propto }_1{\rho }_1\right)}^n_{i;j-1}}{\mathrm{\Delta }\mathrm{y}}\left(v_1\right)}^n_{i;j-{1}/{2}}\right]\mathrm{\Delta }\mathrm{x}=$ $\ {\left[\mathrm{-}\mathrm{\Gamma }e+\mathrm{\Gamma }c+{\dot{M}}_{1\ in}\right]}^n_{i;j}\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}$

Is that correct ?

I read also that we can do this approximation with the UPWIND scheme but I am not confident to use this scheme because it is based on the value of the velocity; for example:

${\left({\propto }_1{\rho }_1\right)}^n_{i+1/2;j}=$ ${\left({\propto }_1{\rho }_1\right)}^n_{i;j}$ if ${\left(u_1\right)}^n_{i+1/2;j}>0$

OR

${\left({\propto }_1{\rho }_1\right)}^n_{i+1/2;j}=$ ${\left({\propto }_1{\rho }_1\right)}^n_{i+1;j}$ if ${\left(u_1\right)}^n_{i+1/2;j}<0$

But I have also make the test for ${\left(u_1\right)}^n_{i-1/2;j}$ ${\left(v_1\right)}^n_{i;j+1/2}$ and ${\left(v_1\right)}^n_{i;j-1/2}$ so the fill in of my matrix coefficients become very tough ...

What is the best option?

FMDenaro · April 16, 2016, 14:32

Quote:

Originally Posted by kiwiguigou

So you mean that for the terms ${\left({\propto }_1{\rho }_1\right)}^n_{i+1/2;j}$ I can make the approximation $\frac{{\left({\propto }_1{\rho }_1u_1\right)}^n_{i+1;j}-{\left({\propto }_1{\rho }_1u_1\right)}^n_{i;j}}{\mathrm{\Delta }\mathrm{x}}$ ?

not at all ... you have a convective flux at one section it can not be discretized as first derivative!
You can use zeroth-order polynomial in a first order upwind reconstruction or a linear polynomial for a second order reconstruction

kiwiguigou · April 16, 2016, 15:57

Understood I can't use a first derivative solution.
But I am not sure to fully understand what you mean by "linear polynomial for a second order reconstruction". Could you give me an example ?

Thank you.

April 15, 2016, 15:34	Discretization of mass conservation equation in a 2D staggered grid	#1
kiwiguigou New Member kiwi Join Date: Apr 2016 Posts: 3 Rep Power: 8	Hi everybody, As mentioned in the title I would like to solve the mass conservation to simulate the boiling phenomenon into a tube bundle. I am facing some issues to discretize these equations into a staggered grid after to have applied an integration into a finite volume. Accordingly to the classical staggered grid: the flow properties are content in the control volume whereas the velocity information is on the bondaries of this control volume. Here is a representation of the staggered grid I use (for the notations) My governing equation is: $\frac{\delta \left({\propto }_k{\rho }_k\right)}{\delta t}+\frac{\delta \left({\propto }_k{\rho }_ku_k\right)}{\delta x}+\frac{\delta \left({\propto }_k{\rho }_ku_k\right)}{\delta y}={\left(-1\right)}^k\left(\mathrm{\Gamma }e-\mathrm{\Gamma }c\right)+{\dot{M}}_{k\ in}$ Where the subcript k is for the phase type (1 for liquid; 2 for gas) $\mathrm{\Gamma }c\$ and $\mathrm{\Gamma }e\$ are the evaporation and condensation rate defined as: $\mathrm{\Gamma }e=\frac{{\propto }_1{\rho }_1}{{\tau }_e}.\frac{\left(h_1-h^{'}\right)}{\left(h^{''}-h^{'}\right)}$ $\mathrm{\Gamma }c=\frac{{\propto }_1{\rho }_1}{{\tau }_c}.\frac{\left(h^{'}-h_1\right)}{\left(h^{''}-h^{'}\right)}$ Where H" is the staturated enthalpy of the gas, H' is the saturated enthalpy of the liquid and $\tau$ is the relaxation time. First question: does someone have a definition to calculate that relaxation time? After having integrated the mass conservation equation for the control volume I;J, I get: $\left[{\left({\propto }_1{\rho }_1\right)}^n_{I;J}-{\left({\propto }_1{\rho }_1\right)}^{n-1}_{I;J}\right]\frac{\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}}{\mathrm{\Delta }\mathrm{t}}+\left[{\left({\propto }_1{\rho }_1u_1\right)}^n_{i+1;j}-{\left({\propto }_1{\rho }_1u_1\right)}^n_{i;j}\right]$ $\mathrm{\Delta }\mathrm{y}+\left[{\left({\propto }_1{\rho }_1v_1\right)}^n_{i;j+1}-{\left({\propto }_1{\rho }_1v_1\right)}^n_{i;j}\right]\mathrm{\Delta }\mathrm{x}=$ $\ {\left[\mathrm{-}\mathrm{\Gamma }e+\mathrm{\Gamma }c+{\dot{M}}_{1\ in}\right]}^n_{I;J}\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}$ By making the assumption that the velocity field is known, I can resolve this equation and find the void fraction and the density. My issue arrives now: I don't know how to approximate the terms of void fraction and density which are at the bondaries of the control volume because the information is not stocked there (ex: ${\left({\propto }_1{\rho }_1\right)}^n_{i+1;j}$ ) If someone can help me on this I will be very grateful. In advance, thank you. Kiwi

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April 15, 2016, 17:35		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,292 Rep Power: 67	your grid arrangement is not totally clear to me from the picture... I immagine that rho(i,j), u(i,j), v(i,j) are staggered each other, right? For example, if rho(i,j) is at the position x(i),y(j), then u(i,j) is at x(i)+dx/2,y(j) and v(i,j) is at x(i),y(j)+dy/2? If it is true, the integration for rho is for the CV centred at x(i),y(j).

April 16, 2016, 07:26		#3
kiwiguigou New Member kiwi Join Date: Apr 2016 Posts: 3 Rep Power: 8	Hi FMdenaro and thank you for your answer. Yes you are right; if we use the notation (i,j) (i+1/2) and (j+1/2) the equation can consequenlty be written as: $\left[{\left({\propto }_1{\rho }_1\right)}^n_{i;j}-{\left({\propto }_1{\rho }_1\right)}^{n-1}_{i;j}\right]\frac{\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}}{\mathrm{\Delta }\mathrm{t}}+\left[{\left({\propto }_1{\rho }_1u_1\right)}^n_{i+1/2;j}-{\left({\propto }_1{\rho }_1u_1\right)}^n_{i-1/2;j}\right]$ $\mathrm{\Delta }\mathrm{y}+\left[{\left({\propto }_1{\rho }_1v_1\right)}^n_{i;j+1/2}-{\left({\propto }_1{\rho }_1v_1\right)}^n_{i;j-1/2}\right]\mathrm{\Delta }\mathrm{x}=$ $\ {\left[\mathrm{-}\mathrm{\Gamma }e+\mathrm{\Gamma }c+{\dot{M}}_{1\ in}\right]}^n_{i;j}\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}$ Is it more clear for you with that writing? So you mean that for the terms ${\left({\propto }_1{\rho }_1\right)}^n_{i+1/2;j}$ I can make the approximation $\frac{{\left({\propto }_1{\rho }_1u_1\right)}^n_{i+1;j}-{\left({\propto }_1{\rho }_1u_1\right)}^n_{i;j}}{\mathrm{\Delta }\mathrm{x}}$ ? So the equation would become: $\left[{\left({\propto }_1{\rho }_1\right)}^n_{i;j}-{\left({\propto }_1{\rho }_1\right)}^{n-1}_{i;j}\right]\frac{\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}}{\mathrm{\Delta }\mathrm{t}}+$ $\left[{\frac{{\left({\propto }_1{\rho }_1\right)}^n_{i+1;j}-{\left({\propto }_1{\rho }_1\right)}^n_{i;j}}{\mathrm{\Delta }\mathrm{x}}\left(u_1\right)}^n_{i+{1}/{2};j}-{\frac{{\left({\propto }_1{\rho }_1\right)}^n_{i;j}-{\left({\propto }_1{\rho }_1\right)}^n_{i-1;j}}{\mathrm{\Delta }\mathrm{x}}\left(u_1\right)}^n_{i-{1}/{2};j}\right]\mathrm{\Delta }\mathrm{y}+$ $\left[{\frac{{\left({\propto }_1{\rho }_1\right)}^n_{i;j+1}-{\left({\propto }_1{\rho }_1\right)}^n_{i;j}}{\mathrm{\Delta }\mathrm{y}}\left(v_1\right)}^n_{i;j+{1}/{2}}-{\frac{{\left({\propto }_1{\rho }_1\right)}^n_{i;j}-{\left({\propto }_1{\rho }_1\right)}^n_{i;j-1}}{\mathrm{\Delta }\mathrm{y}}\left(v_1\right)}^n_{i;j-{1}/{2}}\right]\mathrm{\Delta }\mathrm{x}=$ $\ {\left[\mathrm{-}\mathrm{\Gamma }e+\mathrm{\Gamma }c+{\dot{M}}_{1\ in}\right]}^n_{i;j}\mathrm{\Delta }\mathrm{x}\mathrm{\Delta }\mathrm{y}$ Is that correct ? I read also that we can do this approximation with the UPWIND scheme but I am not confident to use this scheme because it is based on the value of the velocity; for example: ${\left({\propto }_1{\rho }_1\right)}^n_{i+1/2;j}=$ ${\left({\propto }_1{\rho }_1\right)}^n_{i;j}$ if ${\left(u_1\right)}^n_{i+1/2;j}>0$ OR ${\left({\propto }_1{\rho }_1\right)}^n_{i+1/2;j}=$ ${\left({\propto }_1{\rho }_1\right)}^n_{i+1;j}$ if ${\left(u_1\right)}^n_{i+1/2;j}<0$ But I have also make the test for ${\left(u_1\right)}^n_{i-1/2;j}$ ${\left(v_1\right)}^n_{i;j+1/2}$ and ${\left(v_1\right)}^n_{i;j-1/2}$ so the fill in of my matrix coefficients become very tough ... What is the best option?

April 16, 2016, 15:57		#5
kiwiguigou New Member kiwi Join Date: Apr 2016 Posts: 3 Rep Power: 8	Understood I can't use a first derivative solution. But I am not sure to fully understand what you mean by "linear polynomial for a second order reconstruction". Could you give me an example ? Thank you.