How to discretize derivative Source Term??

siddiquesil · September 30, 2019, 16:27

I am trying to solve the Navier Stokes Equation (2D) for Spherical Co-ordinates using FVM. I have come across a source term in derivative form.

My souce term is $\frac{2}{r^{2}}\frac{\partial }{\partial r}\left(\mu r^{2} \frac{V_{r}}{r} \right)$

Now after integrating the above term along with north&south faces of the control volume, we can write it as--

$2\left[ \left(\mu r^2 \frac{V_r}{r} \right)_n - \left(\mu r^2 \frac{V_r}{r} \right)_s\right]\triangle\theta sin\theta_p$

which can be further simplified as
$2\left[\mu_{n}\left(area\right)_n\left(\frac{V_r}{r}\right)_{n}- \mu_{s}\left(area\right)_s\left(\frac{V_r}{r}\right)_{s}\right]$

where
$\left(area\right)_n = r_n^2\triangle\theta sin\theta_{P}$
$\left(area\right)_s = r_s^2\triangle\theta sin\theta_{P}$

My questions are.
1. Should I interpolate the velocity to the main grid point from north and south faces?
2. After interpolating, should I used the old values for the source term?
3. Is there any other way to handle this kind of source term?
4. For source term linearization, as per Patankar formulation S=Sc+Sp.U
Can I have source terms in other grid points also?

I mean to say, as like S= Sc+Sp.U
Can I have S=Sc+Sw.U, S=Sc+Sn.U, S=Sc+Ss.U, S=Sc+Se.U ?

5. Can you please suggest me any good book/reference for my query?

Regards.
Numan Siddique
siddiquesil@gmail.com

FMDenaro · September 30, 2019, 17:04

Quote:

Originally Posted by siddiquesil

I am trying to solve the Navier Stokes Equation (2D) for Spherical Co-ordinates using FVM. I have come across a source term in derivative form.

My souce term is $\frac{2}{r^{2}}\frac{\partial }{\partial r}\left(\mu r^{2} \frac{V_{r}}{r} \right)$

Now after integrating the above term along with north&south faces of the control volume, we can write it as--

$2\left[ \left(\mu r^2 \frac{V_r}{r} \right)_n - \left(\mu r^2 \frac{V_r}{r} \right)_s\right]\triangle\theta sin\theta_p$

which can be further simplified as
$2\left[\mu_{n}\left(area\right)_n\left(\frac{V_r}{r}\right)_{n}- \mu_{s}\left(area\right)_s\left(\frac{V_r}{r}\right)_{s}\right]$

where
$\left(area\right)_n = r_n^2\triangle\theta sin\theta_{P}$
$\left(area\right)_s = r_s^2\triangle\theta sin\theta_{P}$

My questions are.
1. Should I interpolate the velocity to the main grid point from north and south faces?
2. After interpolating, should I used the old values for the source term?
3. Is there any other way to handle this kind of source term?
4. For source term linearization, as per Patankar formulation S=Sc+Sp.U
Can I have source terms in other grid points also?

I mean to say, as like S= Sc+Sp.U
Can I have S=Sc+Sw.U, S=Sc+Sn.U, S=Sc+Ss.U, S=Sc+Se.U ?

5. Can you please suggest me any good book/reference for my query?

Regards.
Numan Siddique
siddiquesil@gmail.com

What is wrong with your approach is that you are writing the equations for a finite difference scheme...conversely, you must start from the general conservative form and write the integral of the fluxes.

Have a look for example to this paper https://arxiv.org/abs/1404.0537

siddiquesil · September 30, 2019, 19:26

Thank you for your reply.

I have discretized the N.S equation for spherical coordinate using Finite volume method. Here are the full details of the discretization scheme where I have used the flux term.

Navier Equation for spherical coordinate in r-direction is given as

$\frac{\partial \left(\rho V_r\right)}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(\rho r^2 V_r^2\right)+\frac{1}{r sin\theta}\frac{\partial }{\partial \theta}\left(\rho sin\theta V_r V_\theta\right)-\frac{\rho V_\theta^2}{r}=$
$-\frac{\partial p}{\partial r}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(\mu r^2 \frac{\partial V_r}{\partial r}\right)+\frac{1}{r^2 sin\theta}\frac{\partial }{\partial \theta}\left(\mu sin\theta \frac{\partial V_r}{\partial \theta}\right)+\frac{2}{r^2}\frac{\partial }{\partial r}\left(\mu r^2 \frac{V_r}{r}\right)$

After rearranging the convection-diffusion term, the above equation can be written as

$\frac{\partial \left(\rho V_r\right)}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(\rho r^2 V_r^2 - \mu r^2 \frac{\partial V_r}{\partial r}\right)+\frac{1}{r sin\theta}\frac{\partial }{\partial \theta}\left(\rho sin\theta V_r V_\theta - \mu sin\theta \frac{\partial V_r}{r\partial \theta}\right)=$
$-\frac{\partial p}{\partial r}+\frac{\rho V_\theta^2}{r}+\frac{2}{r^2}\frac{\partial }{\partial r}\left(\mu r^2 \frac{V_r}{r}\right)$

which can be written by using flux term

$\frac{\partial \left(\rho V_r\right)}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left( r^2 J_r\right)+\frac{1}{r sin\theta}\frac{\partial }{\partial \theta}\left( sin\theta J_\theta\right)=-\frac{\partial p}{\partial r}+\frac{\rho V_\theta^2}{r}+\frac{2}{r^2}\frac{\partial }{\partial r}\left(\mu r^2 \frac{V_r}{r}\right)$

where the fluxes are defined as
$J_r=\left(\rho V_r^2 - \mu \frac{\partial V_r}{\partial r}\right)$ , $J_\theta=\left(\rho V_r V_\theta - \mu \frac{\partial V_r}{r\partial \theta}\right)$

In the above equation, all the term of the right-hand side is source terms.
First two-term of the right-hand of the above equation can be easily found out. Pressure term is hand via the SIMPLER algorithm.

My doubt is about the 3rd source term of the right-hand side of the equation.

siddiquesil · September 30, 2019, 19:36

after discretization of the above equation, the final form can be written as.

$A_n\left(J_r\right)_n-A_s\left(J_r\right)_s+A_w\left(J_\theta\right)_w-A_e\left(J_\theta\right)_e=$ $A_e(P_E-P_W)+\left(\frac{\rho V_\theta^2}{r}\right)_P \triangle V+2\left[A_n\left(\frac{V_r}{r}\right)_n-A_s\left(\frac{V_r}{r}\right)_s\right]$

Where,

= area of the north face

= area of the south face

=area of the west face

=area of the east face
$\triangle V$ = volume.

My questions are only about the last term.
1. How to handle the last term of the above equation?
2. Should I interpolate the value of north&south faces to the main grid point for the last term?
3. Can I write the last term as S=Sc+Sn.V & S=Sc+Ss.V as like S=Sc+Sp.V?

FMDenaro · October 1, 2019, 08:18

Actually, there is no source term in the momentum equation. A source term is something that must be added explicitly as a consequence of a physical external forcing.
From what you wrote I see you are not focusing the fact that in a FVM the fluxes must be written to ensure conservation. You haven't read the paper I linked.

siddiquesil · October 1, 2019, 20:54

Thank you very much. Finally, I have understood my mistakes as there would not any extra terms as a source in the momentum equation apart from the pressure, and centrifugal forces in my case. I would like to read the reference provided by you. Thank you again.

regards:
Numan Siddique Mazumder.
India

September 30, 2019, 16:27	How to discretize derivative Source Term??	#1
siddiquesil Member Numan Mazumder Join Date: Jan 2019 Location: India Posts: 35 Rep Power: 7	I am trying to solve the Navier Stokes Equation (2D) for Spherical Co-ordinates using FVM. I have come across a source term in derivative form. My souce term is $\frac{2}{r^{2}}\frac{\partial }{\partial r}\left(\mu r^{2} \frac{V_{r}}{r} \right)$ Now after integrating the above term along with north&south faces of the control volume, we can write it as-- $2\left[ \left(\mu r^2 \frac{V_r}{r} \right)_n - \left(\mu r^2 \frac{V_r}{r} \right)_s\right]\triangle\theta sin\theta_p$ which can be further simplified as $2\left[\mu_{n}\left(area\right)_n\left(\frac{V_r}{r}\right)_{n}- \mu_{s}\left(area\right)_s\left(\frac{V_r}{r}\right)_{s}\right]$ where $\left(area\right)_n = r_n^2\triangle\theta sin\theta_{P}$ $\left(area\right)_s = r_s^2\triangle\theta sin\theta_{P}$ My questions are. 1. Should I interpolate the velocity to the main grid point from north and south faces? 2. After interpolating, should I used the old values for the source term? 3. Is there any other way to handle this kind of source term? 4. For source term linearization, as per Patankar formulation S=Sc+Sp.U Can I have source terms in other grid points also? I mean to say, as like S= Sc+Sp.U Can I have S=Sc+Sw.U, S=Sc+Sn.U, S=Sc+Ss.U, S=Sc+Se.U ? 5. Can you please suggest me any good book/reference for my query? Regards. Numan Siddique siddiquesil@gmail.com

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September 30, 2019, 19:26		#3
siddiquesil Member Numan Mazumder Join Date: Jan 2019 Location: India Posts: 35 Rep Power: 7	Thank you for your reply. I have discretized the N.S equation for spherical coordinate using Finite volume method. Here are the full details of the discretization scheme where I have used the flux term. Navier Equation for spherical coordinate in r-direction is given as $\frac{\partial \left(\rho V_r\right)}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(\rho r^2 V_r^2\right)+\frac{1}{r sin\theta}\frac{\partial }{\partial \theta}\left(\rho sin\theta V_r V_\theta\right)-\frac{\rho V_\theta^2}{r}=$ $-\frac{\partial p}{\partial r}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(\mu r^2 \frac{\partial V_r}{\partial r}\right)+\frac{1}{r^2 sin\theta}\frac{\partial }{\partial \theta}\left(\mu sin\theta \frac{\partial V_r}{\partial \theta}\right)+\frac{2}{r^2}\frac{\partial }{\partial r}\left(\mu r^2 \frac{V_r}{r}\right)$ After rearranging the convection-diffusion term, the above equation can be written as $\frac{\partial \left(\rho V_r\right)}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(\rho r^2 V_r^2 - \mu r^2 \frac{\partial V_r}{\partial r}\right)+\frac{1}{r sin\theta}\frac{\partial }{\partial \theta}\left(\rho sin\theta V_r V_\theta - \mu sin\theta \frac{\partial V_r}{r\partial \theta}\right)=$ $-\frac{\partial p}{\partial r}+\frac{\rho V_\theta^2}{r}+\frac{2}{r^2}\frac{\partial }{\partial r}\left(\mu r^2 \frac{V_r}{r}\right)$ which can be written by using flux term $\frac{\partial \left(\rho V_r\right)}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left( r^2 J_r\right)+\frac{1}{r sin\theta}\frac{\partial }{\partial \theta}\left( sin\theta J_\theta\right)=-\frac{\partial p}{\partial r}+\frac{\rho V_\theta^2}{r}+\frac{2}{r^2}\frac{\partial }{\partial r}\left(\mu r^2 \frac{V_r}{r}\right)$ where the fluxes are defined as $J_r=\left(\rho V_r^2 - \mu \frac{\partial V_r}{\partial r}\right)$ , $J_\theta=\left(\rho V_r V_\theta - \mu \frac{\partial V_r}{r\partial \theta}\right)$ In the above equation, all the term of the right-hand side is source terms. First two-term of the right-hand of the above equation can be easily found out. Pressure term is hand via the SIMPLER algorithm. My doubt is about the 3rd source term of the right-hand side of the equation.

September 30, 2019, 19:36		#4
siddiquesil Member Numan Mazumder Join Date: Jan 2019 Location: India Posts: 35 Rep Power: 7	after discretization of the above equation, the final form can be written as. $A_n\left(J_r\right)_n-A_s\left(J_r\right)_s+A_w\left(J_\theta\right)_w-A_e\left(J_\theta\right)_e=$ $A_e(P_E-P_W)+\left(\frac{\rho V_\theta^2}{r}\right)_P \triangle V+2\left[A_n\left(\frac{V_r}{r}\right)_n-A_s\left(\frac{V_r}{r}\right)_s\right]$ Where, = area of the north face = area of the south face =area of the west face =area of the east face $\triangle V$ = volume. My questions are only about the last term. 1. How to handle the last term of the above equation? 2. Should I interpolate the value of north&south faces to the main grid point for the last term? 3. Can I write the last term as S=Sc+Sn.V & S=Sc+Ss.V as like S=Sc+Sp.V?

October 1, 2019, 08:18		#5
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,897 Rep Power: 73	Actually, there is no source term in the momentum equation. A source term is something that must be added explicitly as a consequence of a physical external forcing. From what you wrote I see you are not focusing the fact that in a FVM the fluxes must be written to ensure conservation. You haven't read the paper I linked.

October 1, 2019, 20:54		#6
siddiquesil Member Numan Mazumder Join Date: Jan 2019 Location: India Posts: 35 Rep Power: 7	Thank you very much. Finally, I have understood my mistakes as there would not any extra terms as a source in the momentum equation apart from the pressure, and centrifugal forces in my case. I would like to read the reference provided by you. Thank you again. regards: Numan Siddique Mazumder. India