Navier-Stokes equation in spherical coordinate

optimux · February 16, 2022, 03:08

Hi All,

I need to solve Navier-Stokes equation in spherical coordinate. Prof. Batchelor gave mass and momentum equations directly without derivation in his book "An introduction to Fluid Mechanics", 1967. Besides, some show a coordinate transformation from Cartesian, which is not clear from a physical basis. So I want a complete derivation from basic conservation laws. Could someone help? Thank you so much.

Best,
Mingming

Gerry Kan · February 16, 2022, 05:08

Dear Mingming:

If you are looking for a direct transformation between the Cartesian equations to the spherical coordinates, look up something called "Lamé coefficients." This should simplify your work.

Otherwise, if you are looking for a direct derivation of these equations in spherical coordinates, you might want to start with mass conservation (continuity) and try to derive it yourself and then infer (not repeat) this for the momentum equations.

Hope that helps, Gerry.

FMDenaro · February 16, 2022, 05:13

Quote:

Originally Posted by optimux

Hi All,

I need to solve Navier-Stokes equation in spherical coordinate. Prof. Batchelor gave mass and momentum equations directly without derivation in his book "An introduction to Fluid Mechanics", 1967. Besides, some show a coordinate transformation from Cartesian, which is not clear from a physical basis. So I want a complete derivation from basic conservation laws. Could someone help? Thank you so much.

Best,
Mingming

That means you want to write the Reynolds transport equation in spherical coordinates and then mass, momentum and energy?

optimux · February 16, 2022, 05:25

Quote:

Originally Posted by Gerry Kan

Dear Mingming:

If you are looking for a direct transformation between the Cartesian equations to the spherical coordinates, look up something called "Lamé coefficients." This should simplify your work.

Otherwise, if you are looking for a direct derivation of these equations in spherical coordinates, you might want to start with mass conservation (continuity) and try to derive it yourself and then infer (not repeat) this for the momentum equations.

Hope that helps, Gerry.

Hi Gerry,

Yes, I derived continuity equation by myself from mass conservation, but failed for the momentum equations. There must be something missing that I have to dig into.

Thank you so much.

Best,
Mingming

optimux · February 16, 2022, 05:35

Quote:

Originally Posted by FMDenaro

That means you want to write the Reynolds transport equation in spherical coordinates and then mass, momentum and energy?

Hey Denaro,

Yes, mass, momentum, energy and chemical species conservation equations for incompressible flows. I have derived them from scratch in Cartesian coordinate, with porous media included. But spherical coordinate is quite different. Could you please give me some hint?

Thanks,
Mingming

FMDenaro · February 16, 2022, 05:46

Quote:

Originally Posted by optimux

Hey Denaro,

Yes, mass, momentum, energy and chemical species conservation equations for incompressible flows. I have derived them from scratch in Cartesian coordinate, with porous media included. But spherical coordinate is quite different. Could you please give me some hint?

Thanks,
Mingming

The NSE in spherical coordinates are detailed in many textbooks.

I suggest to start from the simple polar representation (without porous effects) to check if you are doing correctly your derivation.
You need to express the nabla and integral operators in sperical coordinates and then apply the inner product to the momentum variable.
Maybe you could write the point in which your derivation failed.

optimux · February 16, 2022, 06:10

Quote:

Originally Posted by FMDenaro

The NSE in spherical coordinates are detailed in many textbooks.

I suggest to start from the simple polar representation (without porous effects) to check if you are doing correctly your derivation.
You need to express the nabla and integral operators in sperical coordinates and then apply the inner product to the momentum variable.
Maybe you could write the point in which your derivation failed.

Hi Denaro,

I probably missed unit vectors relationship but I can not tell exactly. Yes, I tried cylindrical coordinate without porous media but not complete yet.

BTW, would you please name a book which has a complete derivation? Thank you.

Best,
Mingming

FMDenaro · February 16, 2022, 06:50

Quote:

Originally Posted by optimux

Hi Denaro,

I probably missed unit vectors relationship but I can not tell exactly. Yes, I tried cylindrical coordinate without porous media but not complete yet.

BTW, would you please name a book which has a complete derivation? Thank you.

Best,
Mingming

post your notes and we can check ....
Have a look to the dedicated appendix in the textbook of Kundu.

optimux · February 16, 2022, 07:15

Quote:

Originally Posted by FMDenaro

post your notes and we can check ....
Have a look to the dedicated appendix in the textbook of Kundu.

Hi Denaro,

Thanks a lot! I'll check out that book first and translate my notes. Thanks for your time.

Best,
Mingming

Gerry Kan · February 16, 2022, 07:23

I remember this being in my course notes from some decades ago. The direct derivation of the momentum equations in the spherical coordinates are not so straightforward.

What you can do, is to assume a set of transformation coefficients (i.e., Lamé coefficients). Leave them "as is" (i.e., as h1, h2, and h3) until the very end, since you are deriving differential equations. These coefficients are conservative so the advantage is that you simply have to carry them in your derivation and then substitute them when you have the governing equations in basic form, effectively treating the derivation as Cartesian for the most part.

I imagine this is also the same approach as what is shown in Batchelor's text. What these coefficients are you can infer from the continuity equation, which you have already done.

Also, how are you modelling porous media? With Darcy / Forchheimer relations, or explicitly (i.e., porous interface as walls)?

Gerry.

optimux · February 16, 2022, 08:10

Quote:

Originally Posted by Gerry Kan

I remember this being in my course notes from some decades ago. The direct derivation of the momentum equations in the spherical coordinates are not so straightforward.

What you can do, is to assume a set of transformation coefficients (i.e., Lamé coefficients). Leave them "as is" (i.e., as h1, h2, and h3) until the very end, since you are deriving differential equations. These coefficients are conservative so the advantage is that you simply have to carry them in your derivation and then substitute them when you have the governing equations in basic form, effectively treating the derivation as Cartesian for the most part.

I imagine this is also the same approach as what is shown in Batchelor's text. What these coefficients are you can infer from the continuity equation, which you have already done.

Also, how are you modelling porous media? With Darcy / Forchheimer relations, or explicitly (i.e., porous interface as walls)?

Gerry.

Hi Gerry,

Thanks for your suggestions. I use Darcy's law to model resistence of porous media.

Best,
Mingming

Gerry Kan · February 16, 2022, 10:20

Quote:

Originally Posted by optimux

Thanks for your suggestions. I use Darcy's law to model resistence of porous media.

Do keep in mind that Darcy's law is an empirical loss term. For derive the resistance in all three spherical directions, assume the general form:

$u_k = -\left(\frac{\kappa}{\mu}\right)\frac{1}{h_k}\nabla_k{p}$

where $\kappa$ is the permeability, $\mu$ is the dynamic viscosity,

is the individual orthogonal component $r, \phi, \theta$ , and

are the Lamé coefficients for each of the component.

What you will have should be:

$k_r = 1, \quad k_\phi = r\sin(\theta), \quad k_\theta = r$ .

If you need to be convinced for the spherical coordinates, you can derive the Lamé coefficients from mass conservation alone. The corresponding differentials ( $\nabla, \nabla\cdotp, \nabla\times, \nabla^2$ ) and substitute this into your momentum equations to save you work in deriving everything, since it will get very messy for spherical coordinates.

Gerry.

optimux · February 16, 2022, 12:09

Quote:

Originally Posted by Gerry Kan

Do keep in mind that Darcy's law is an empirical loss term. For derive the resistance in all three spherical directions, assume the general form:

$u_k = -\left(\frac{\kappa}{\mu}\right)\frac{1}{h_k}\nabla_k{p}$

where $\kappa$ is the permeability, $\mu$ is the dynamic viscosity,

is the individual orthogonal component $r, \phi, \theta$ , and

are the Lamé coefficients for each of the component.

What you will have should be:

$k_r = 1, \quad k_\phi = r\sin(\theta), \quad k_\theta = r$ .

If you need to be convinced for the spherical coordinates, you can derive the Lamé coefficients from mass conservation alone. The corresponding differentials ( $\nabla, \nabla\cdotp, \nabla\times, \nabla^2$ ) and substitute this into your momentum equations to save you work in deriving everything, since it will get very messy for spherical coordinates.

Gerry.

Hi Gerry,

It seems that spherical coordinate is not friendly. I'll keep this in mind. Thank you so much.

Best,
Mingming

LuckyTran · February 16, 2022, 13:38

In my opinion, this is best done by deriving the vector form of the Navier-Stokes equations using divergence operators (this is also the meta). Then you can freely go into any coordinate system you want. Coordinates are just representations of vectors, the underlying physics remains the same.

sonmezroy · February 24, 2022, 05:07

I derived continuity equation by myself from mass conservation, but failed for the momentum equations. shareit downloading vidmate 2014

optimux · February 26, 2022, 04:14

Quote:

Originally Posted by LuckyTran

In my opinion, this is best done by deriving the vector form of the Navier-Stokes equations using divergence operators (this is also the meta). Then you can freely go into any coordinate system you want. Coordinates are just representations of vectors, the underlying physics remains the same.

Hi Tran,

I followed your suggestion and succeeded. At this moment, only mass and momentum equations are derived without porous media, species and energy equations are the next step.

Thank you very much.

Best,
Mingming

optimux · February 26, 2022, 04:19

Quote:

Originally Posted by sonmezroy

I derived continuity equation by myself from mass conservation, but failed for the momentum equations.

Hi Sonmezroy,

How did you work out mass conservation equation? I'd like to share my notes but it's not in English. You can email me at zmmhst0819@gmail.com.

Best,
Mingming

optimux · February 26, 2022, 04:25

Dear CFDers,

I have derived mass and momentum equations in sperical coordinates by hand, to have a deeper understanding of variables in spherical coordiantes. Anyone interested in could contact me at zmmhst0819@gmail.com.

Cheers,
Mingming

optimux · March 9, 2022, 11:52

Quote:

Originally Posted by LuckyTran

In my opinion, this is best done by deriving the vector form of the Navier-Stokes equations using divergence operators (this is also the meta). Then you can freely go into any coordinate system you want. Coordinates are just representations of vectors, the underlying physics remains the same.

Hi Tran,

I recently found a paper that uses finite difference method to solve NS equations in spherical coordinates. It provides an interesting pseudo-velocity method to circumvent singularity problem. It's "A finite difference scheme for three-dimensional incompressible fows in spherical coordinates" by Luca Santelli.

Best,
Mingming

unogamerules · July 28, 2022, 11:08

Yes, I derived continuity equation by myself from mass conservation, but failed for the momentum equations. There must be something missing that I have to dig into.

February 16, 2022, 03:08	Navier-Stokes equation in spherical coordinate	#1
optimux Member Mingming Zhang Join Date: Dec 2019 Posts: 35 Rep Power: 6	Hi All, I need to solve Navier-Stokes equation in spherical coordinate. Prof. Batchelor gave mass and momentum equations directly without derivation in his book "An introduction to Fluid Mechanics", 1967. Besides, some show a coordinate transformation from Cartesian, which is not clear from a physical basis. So I want a complete derivation from basic conservation laws. Could someone help? Thank you so much. Best, Mingming

February 24, 2022, 05:07		#15
sonmezroy New Member cartlon flores Join Date: Feb 2022 Posts: 1 Rep Power: 0	I derived continuity equation by myself from mass conservation, but failed for the momentum equations. shareit downloading vidmate 2014 Last edited by sonmezroy; February 26, 2022 at 05:09.

February 26, 2022, 04:25	Mass+momentum equation in spherical coordinates derivation	#18
optimux Member Mingming Zhang Join Date: Dec 2019 Posts: 35 Rep Power: 6	Dear CFDers, I have derived mass and momentum equations in sperical coordinates by hand, to have a deeper understanding of variables in spherical coordiantes. Anyone interested in could contact me at zmmhst0819@gmail.com. Cheers, Mingming

July 28, 2022, 11:08		#20
unogamerules New Member Uno Join Date: Jul 2022 Posts: 1 Rep Power: 0	Yes, I derived continuity equation by myself from mass conservation, but failed for the momentum equations. There must be something missing that I have to dig into. __________________ UNO blank card rules

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February 16, 2022, 05:08		#2
Gerry Kan Senior Member Gerry Kan Join Date: May 2016 Posts: 373 Rep Power: 11	Dear Mingming: If you are looking for a direct transformation between the Cartesian equations to the spherical coordinates, look up something called "Lamé coefficients." This should simplify your work. Otherwise, if you are looking for a direct derivation of these equations in spherical coordinates, you might want to start with mass conservation (continuity) and try to derive it yourself and then infer (not repeat) this for the momentum equations. Hope that helps, Gerry.

February 16, 2022, 07:23		#10
Gerry Kan Senior Member Gerry Kan Join Date: May 2016 Posts: 373 Rep Power: 11	I remember this being in my course notes from some decades ago. The direct derivation of the momentum equations in the spherical coordinates are not so straightforward. What you can do, is to assume a set of transformation coefficients (i.e., Lamé coefficients). Leave them "as is" (i.e., as h1, h2, and h3) until the very end, since you are deriving differential equations. These coefficients are conservative so the advantage is that you simply have to carry them in your derivation and then substitute them when you have the governing equations in basic form, effectively treating the derivation as Cartesian for the most part. I imagine this is also the same approach as what is shown in Batchelor's text. What these coefficients are you can infer from the continuity equation, which you have already done. Also, how are you modelling porous media? With Darcy / Forchheimer relations, or explicitly (i.e., porous interface as walls)? Gerry.

February 16, 2022, 13:38		#14
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,753 Rep Power: 66	In my opinion, this is best done by deriving the vector form of the Navier-Stokes equations using divergence operators (this is also the meta). Then you can freely go into any coordinate system you want. Coordinates are just representations of vectors, the underlying physics remains the same.