CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > General Forums > Main CFD Forum

Internal Energy equation in viscous compressible floe equations

Register Blogs Members List Search Today's Posts Mark Forums Read

Reply
 
LinkBack Thread Tools Search this Thread Display Modes
Old   August 17, 2015, 16:34
Default Internal Energy equation in viscous compressible floe equations
  #1
Member
 
Mihir Makwana
Join Date: May 2015
Posts: 79
Rep Power: 11
mihirmakwana6 is on a distinguished road
How can I use FVM to discretize the term

sigma : D


in the internal energy equation

http://i.imgur.com/3RaYm5p.jpg

where

sigma and D are given by

http://i.imgur.com/6K65Lih.jpg


I am not able to expand the term sigma : D

Please help.

Thanks in Advance

- Mihir
mihirmakwana6 is offline   Reply With Quote

Old   August 17, 2015, 18:03
Default
  #2
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,892
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
If you want using a FVM, you have to recast the equation in the divergence form and integrate over a finite volume each term.
Using the expression of sigma, the term D:D you get is a scalar function that appears as volume source
FMDenaro is offline   Reply With Quote

Old   August 18, 2015, 02:21
Default
  #3
Member
 
Mihir Makwana
Join Date: May 2015
Posts: 79
Rep Power: 11
mihirmakwana6 is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
If you want using a FVM, you have to recast the equation in the divergence form and integrate over a finite volume each term.
Using the expression of sigma, the term D:D you get is a scalar function that appears as volume source
sir, if i use tensors then sigma : D gives

http://i.imgur.com/UmzDCrI.jpg

How do i convert this equation to divergence form ?
mihirmakwana6 is offline   Reply With Quote

Old   August 18, 2015, 03:45
Default
  #4
Member
 
Mihir Makwana
Join Date: May 2015
Posts: 79
Rep Power: 11
mihirmakwana6 is on a distinguished road
sir say i split D

then one of the term is

http://i.imgur.com/zkbee0D.jpg


here Tau is sigma

so the 1st term on r.h.s is in divergence form but the second is not
mihirmakwana6 is offline   Reply With Quote

Old   August 18, 2015, 04:13
Default
  #5
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,892
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
no, you do not have all terms in te internal energy equation in divergence form...that respects the physical fact that such energy form does not have a conservative formulation. The term D: D is a pointwise source term, you have to integrate over the local volume in a FVM and discretize the volume integral.
If you want to work with a fully divergence form then you have to adopt the total energy equation
FMDenaro is offline   Reply With Quote

Old   August 18, 2015, 05:58
Default
  #6
Member
 
Mihir Makwana
Join Date: May 2015
Posts: 79
Rep Power: 11
mihirmakwana6 is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
no, you do not have all terms in te internal energy equation in divergence form...that respects the physical fact that such energy form does not have a conservative formulation. The term D: D is a pointwise source term, you have to integrate over the local volume in a FVM and discretize the volume integral.
If you want to work with a fully divergence form then you have to adopt the total energy equation
ok.

1) As i am using FVM, i cannot integrate the second term in RHS of

http://i.imgur.com/zkbee0D.jpg

Right ??

or is there a way I can integrate it over the C.V

2) what is D: D ?
mihirmakwana6 is offline   Reply With Quote

Old   August 18, 2015, 06:02
Default
  #7
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,892
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
you can always integrate each term of the equation over a finite volume...when the term is in divergence form you can apply Gauss and get the surface integral of the fluxes, conversely the integral to discretize remains the volume integral (see for example the book of Peric & Ferziger).

D: D is nothing else that the double dot product between the symmetric velocity gradient
FMDenaro is offline   Reply With Quote

Old   August 18, 2015, 06:04
Default
  #8
Member
 
Mihir Makwana
Join Date: May 2015
Posts: 79
Rep Power: 11
mihirmakwana6 is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
you can always integrate each term of the equation over a finite volume...when the term is in divergence form you can apply Gauss and get the surface integral of the fluxes, conversely the integral to discretize remains the volume integral (see for example the book of Peric & Ferziger).

D: D is nothing else that the double dot product between the symmetric velocity gradient
1) ok.

2) but i need to find sigma : D and not D : D
mihirmakwana6 is offline   Reply With Quote

Old   August 18, 2015, 06:11
Default
  #9
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,892
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
sigma: D is nothing else that a product by an isotropic tensor (I: D) added with D: D

for brevity I disregarded the coefficients
FMDenaro is offline   Reply With Quote

Reply

Thread Tools Search this Thread
Search this Thread:

Advanced Search
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are Off
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
problem on energy and flow equation solving separately preetam69 FLUENT 2 July 30, 2013 22:08
k-e turbulence model and energy equation Blob Main CFD Forum 0 May 29, 2009 09:35
how to get fluid internal internal energy vbk FLUENT 0 December 17, 2007 02:54
question about energy equation zhou FLUENT 0 February 24, 2004 00:55
Why FVM for high-Re flows? Zhong Lei Main CFD Forum 23 May 14, 1999 14:22


All times are GMT -4. The time now is 18:45.