Orr Sommerfeld equation

thegodfather · March 18, 2012, 01:12

My question is very fundamental.
why, at all do we need to have a separate equation that predicts the growth or decay of perturbations in a steady base flow..? If at all, there are any perturbations in the flow, why can't the resultant motion of fluid be predicted using the usual Navier Stokes equations??

truffaldino · March 18, 2012, 04:27

Because Navier-Stokes equation is too complicated to be solved analytically. This is the main problem of all fluid dynamics. That is why we still know almost nothing exact about turbulence.

thegodfather · March 18, 2012, 06:43

thanks a lot for replying... but as far as i know, even Orr - Sommerfeld equation is a fourth order linear differential equation which cannot be solved analytically for most cases, and various numerical methods have to be resorted to solve it... In that case would it not be better to bear with a little more complexity in terms of better numerical methods and solve the full N-S equations?

truffaldino · March 18, 2012, 06:57

As you said Orr-Sommerfeld is a linear system, while Navier-Stokes is nonlinear and shows chaotic behavior (turbulence) which is difficult analyse even numerically (DNS is done on supercomputers).

In the case of Orr-Somerfeld you have a possibility of qualitative analysys of transition to turbulence. You can see how perturbations amplify etc.

thegodfather · March 18, 2012, 07:04

thanks.. that was really helpful..

March 18, 2012, 01:12	Orr Sommerfeld equation	#1
thegodfather New Member Aniket Sachdeva Join Date: Mar 2012 Posts: 22 Rep Power: 14	My question is very fundamental. why, at all do we need to have a separate equation that predicts the growth or decay of perturbations in a steady base flow..? If at all, there are any perturbations in the flow, why can't the resultant motion of fluid be predicted using the usual Navier Stokes equations??

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March 18, 2012, 04:27		#2
truffaldino Senior Member Join Date: Jan 2011 Posts: 249 Blog Entries: 5 Rep Power: 17	Because Navier-Stokes equation is too complicated to be solved analytically. This is the main problem of all fluid dynamics. That is why we still know almost nothing exact about turbulence.

March 18, 2012, 06:43		#3
thegodfather New Member Aniket Sachdeva Join Date: Mar 2012 Posts: 22 Rep Power: 14	thanks a lot for replying... but as far as i know, even Orr - Sommerfeld equation is a fourth order linear differential equation which cannot be solved analytically for most cases, and various numerical methods have to be resorted to solve it... In that case would it not be better to bear with a little more complexity in terms of better numerical methods and solve the full N-S equations?

March 18, 2012, 06:57		#4
truffaldino Senior Member Join Date: Jan 2011 Posts: 249 Blog Entries: 5 Rep Power: 17	As you said Orr-Sommerfeld is a linear system, while Navier-Stokes is nonlinear and shows chaotic behavior (turbulence) which is difficult analyse even numerically (DNS is done on supercomputers). In the case of Orr-Somerfeld you have a possibility of qualitative analysys of transition to turbulence. You can see how perturbations amplify etc.

March 18, 2012, 07:04		#5
thegodfather New Member Aniket Sachdeva Join Date: Mar 2012 Posts: 22 Rep Power: 14	thanks.. that was really helpful..