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

Elliptic-parabolic/Hyperbolic-parabolic N.S Equations

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

Like Tree2Likes
  • 1 Post By FMDenaro
  • 1 Post By FMDenaro

Reply
 
LinkBack Thread Tools Display Modes
Old   January 4, 2014, 04:38
Default Elliptic-parabolic/Hyperbolic-parabolic N.S Equations
  #1
Senior Member
 
Vino
Join Date: Mar 2013
Location: India
Posts: 106
Rep Power: 4
Vino is on a distinguished road
Hi,

I did not get the clear picture of Elliptic-parabolic system (incompressible) or Hyperbolic- parabolic system(compressible) of N.S equations. Let us say for a 2D case, N.S contains 1 continuity and 2 momentum equations. Among these, which will be elliptic/parabolic/hyperbolic? and these hyperbolic/parabolic with respect to time or space?

Please provide me some material/reference to read...
Vino is offline   Reply With Quote

Old   January 4, 2014, 07:01
Default
  #2
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,600
Rep Power: 23
FMDenaro will become famous soon enough
Quote:
Originally Posted by Vino View Post
Hi,

I did not get the clear picture of Elliptic-parabolic system (incompressible) or Hyperbolic- parabolic system(compressible) of N.S equations. Let us say for a 2D case, N.S contains 1 continuity and 2 momentum equations. Among these, which will be elliptic/parabolic/hyperbolic? and these hyperbolic/parabolic with respect to time or space?


Please provide me some material/reference to read...

The continuity equation is always hyperbolic (compressible or not flow model), the momentum equation is parabolic for unsteady viscous (compressible or not) flows but becomes hyperbolic for non-viscous flows.

The mathematical classification of systems of PDE is a topic of applied mathematics but you can find also in CFD books some references.
Vino likes this.
FMDenaro is offline   Reply With Quote

Old   January 4, 2014, 09:10
Default
  #3
Senior Member
 
Vino
Join Date: Mar 2013
Location: India
Posts: 106
Rep Power: 4
Vino is on a distinguished road
Thank you Filippo Maria Denaro.

I am working on incompressible flow computation, i have read a paper recently which says "elliptic-parabolic nature of Incompressible N.S will become hyperbolic-parabolic when it becomes compressible".

can you please clarify the following.

1) In case of Incompressible unsteady flow, will the continuity equation be hyperbolic?(no time derivative of density)

2) In case of Incompressible steady flow,will the momentum equation be parabolic?

In most cfd books, i could find only the nature of general equations and i could not find anything about N.S system. If u particularly know some books, plz suggest....
Vino is offline   Reply With Quote

Old   January 4, 2014, 09:16
Default
  #4
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,600
Rep Power: 23
FMDenaro will become famous soon enough
1) yes, the continuity equation Div V =0 is hyperbolic, it will be then reorganized in the elliptic pressure equation

2) The Prandtl equations are parabolic in the momentum, while the full NS momentum equation are elliptic in the steady case
Vino likes this.
FMDenaro is offline   Reply With Quote

Old   January 4, 2014, 09:22
Default
  #5
Senior Member
 
Vino
Join Date: Mar 2013
Location: India
Posts: 106
Rep Power: 4
Vino is on a distinguished road
Thank you very much.!!!
Vino is offline   Reply With Quote

Old   January 7, 2014, 04:23
Default
  #6
Senior Member
 
duri
Join Date: May 2010
Posts: 130
Rep Power: 7
duri is on a distinguished road
For system of equation it is very difficult to say its kind. Because all equations are coupled. The best way is to find out the eigen value of the system which says system is elliptic or parabolic.
For NS equation, the eigen values are u+a, u-a, u,u,u. When they are positive definite then it is purely elliptic and when it becomes semi definite the it is parabolic. In case of subsonic flow these eigen values can take zero or positive quantity and so they are elliptic-parabolic system.
duri is offline   Reply With Quote

Old   January 7, 2014, 04:44
Default
  #7
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,600
Rep Power: 23
FMDenaro will become famous soon enough
Quote:
Originally Posted by duri View Post
For system of equation it is very difficult to say its kind. Because all equations are coupled. The best way is to find out the eigen value of the system which says system is elliptic or parabolic.
For NS equation, the eigen values are u+a, u-a, u,u,u. When they are positive definite then it is purely elliptic and when it becomes semi definite the it is parabolic. In case of subsonic flow these eigen values can take zero or positive quantity and so they are elliptic-parabolic system.
Well, u+/-a,u are the eigenvalues for the time-dependent Euler system (hyperbolic both in subsonic and supersonic cases), not for NS ... the mathematical character of the NS depends on steady/unsteady formulations
FMDenaro is offline   Reply With Quote

Old   January 7, 2014, 20:46
Default
  #8
Senior Member
 
Vino
Join Date: Mar 2013
Location: India
Posts: 106
Rep Power: 4
Vino is on a distinguished road
I understood the following.!!!

1) Unsteady Incompressible N.S will be Elliptic-parabolic type
2) Unsteady Compressible N.S will be Hyperbolic-parabolic type
3) Steady N.S (for Incompressible & Compressible) will be pure Elliptic type
Vino is offline   Reply With Quote

Reply

Thread Tools
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 On
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Physics of Elliptic Parabolic and Hyperbolic Equations Renjith Main CFD Forum 1 December 4, 2013 05:04
parabolic nad elliptic scheme shahrbanoo Main CFD Forum 1 April 14, 2013 04:51
IdeasUnvToFoam Bug amp Fix benru OpenFOAM Bugs 42 November 13, 2009 08:59
Type of PDE: Hyperbolic or Parabolic or Elliptic? Guo Main CFD Forum 6 January 26, 2001 11:42
Book on N.S. equations J. McCarthur Main CFD Forum 3 August 24, 1999 15:26


All times are GMT -4. The time now is 14:28.