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January 11, 2006, 01:31 
Inertial or Noninertial

#1 
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Hi..
When we do an unsteady simulation using moving boundaries..like an oscillating cylinder, airfoil etc.. do u actually need to solve the NS equations in Noninertial coordinate system or just be in the Inertial coordinate system. regards, B787 

January 11, 2006, 10:04 
Re: Inertial or Noninertial

#2 
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You can solve it either way. If your coordinate system is not accelerating then it is inertial, otherwise it is noninertial. If you use a noninertial system then you will need to account for that in the form of the NavierStokes equations that you use through the inclusion of terms related to the acceleration of the coordinate system.


January 11, 2006, 11:14 
Re: Inertial or Noninertial

#3 
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For the accelerating reference frame terms, what would happen if the moving reference frame turned through an angle?
diaw... 

January 11, 2006, 22:21 
Re: Inertial or Noninertial

#4 
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I'm not sure quite what you are asking, but any rotating frame of reference is noninertial. So you would need to add in the appropriate terms.


January 11, 2006, 22:47 
Re: Inertial or Noninertial

#5 
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Hi Ag,
It is an interesting thread. How is the moving (translating + rotating) reference frame tied back into the conventional NS derivations? These are typically derived with a moving inertial reference frame referenced back to a Eulerian inertial reference frame. In rigidbody dynamics, the moving reference frame translation & rotation are simply performed (in time) v,abs= v,a + v,ref + w x r,a. How does one transform the spatial components due to rotation of the reference frame  i'=wxi , j'=wxj , k'=wxk? From what I have understood, the typical fluid element is a glorified 'particle' (all properties lumped to a central particle) with a 'squishy outer container' (conservation principle, no mass source) . Thanks for your comments 'Ag'... diaw... 

January 11, 2006, 22:54 
Errata: Inertial or Noninertial

#6 
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Errata: Should read  for rigidbody dynamics
v,abs= v,a + v,rel + w x r,p/a where: v,abs = absolute velocity of particle relative to fixed inertial ref frame; v,a = velocity of vertex 'A' of moving ref frame relative to fixed inertial ref frame; v,rel = velocity of particle relative to moving frame = x'i + y'j + z'k; r,p/a = relative position vector of particle to vertex of moving ref frame. w = absolute angular velocity of moving ref frame diaw... 

January 12, 2006, 09:49 
Re: Errata: Inertial or Noninertial

#7 
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I find it easier to consider the motion of the coordinate system rather than think about fluid particles. If you write your equations in vector form and then differentiate with respect to time, carry the time derivatives of the basis vectors along. These will end up yielding the w x r type terms.


January 12, 2006, 09:49 
Re: Inertial or Noninertial

#8 
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Dear Diaw,
In rigid body dynamics, the rotating frame of refernece introduces the Coriolis component, as you have pointed out. But for fluid flow, the Galilean invariance property is useful, as the N/S needs to be modified in the flux terms, replacing the fluid velocity by the relative (Fluid  Grid) velocity. Obviously, the solver would need other changes, but I believe the complications involved in a noninertial frame would be far more. Hope this helps Regards, Ganesh 

January 12, 2006, 13:08 
Re: Inertial or Noninertial

#9 
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Although it's been a while since I derived the equations for a rotating frame, it's not that complicated. Remember that the effect of a noninertial system is a shift in perspective, regardless of whether you are talking about rigid bodies or fluids.


January 12, 2006, 13:56 
Re: Inertial or Noninertial

#10 
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Dear ag,
The derivation of the equations would not be complicated, but the implementation would not be very straight forward. The advantage of maintaining an inertial frame of reference is that you could still use your code to work with nondeforming meshes with stead/unsteady flow, which would serve as an easy check as to whether the coding is right or not. My comment was therfore more from a coding perspective. Regards, Ganesh 

January 12, 2006, 14:23 
Re: Inertial or Noninertial

#11 
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Actually, I don't think the implementation is that bad. I've never done it myself, but I work with several guys who have. Using a noninertial frame of reference is quite common (or was) in the turbomachinery CFD world. From what little I recall the boundary conditions end up being one of the more complicated pieces (not unlike CFD in an inertial system sometimes).


January 12, 2006, 20:58 
Re: Inertial or Noninertial

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Thanks 'ag' & 'ganesh'  you have both raised some very interesting points.
I would agree that to differentiate the vector form of governing eqn in time & then settle the moving basis frame rotation would be the most direct way to sort things out. Nice approach. The current derivations of the NS eqns basically treat the fluid element a 'flexiblesided' element with all its properties lumped to a mass at its centre, & no mass source addition into the element (conservation of mass)  in other words a particle with a 'squishy' outer frame. The rest is absolutely straightforward  when treated as a particle. Using this kinematics approach, the NS can be derived from first principles in less than 5 minutes. Remember, even if you have a noninertial moving ref frame, it must always eventually be referred back to a fixed inertial ref frame somewhere in the system  otherwise Newtons Laws do not hold (momentum eqn = Newton's 2nd law in each direction). diaw... 

January 15, 2006, 10:12 
Re: Inertial or Noninertial

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The question is interesting but the discussion it kicked off seems tedious and misleading.
The grid motion is taken into account in the formulation of the fluxes through cell boundaries. Your coordinate system typically is not affected, i.e. you describe moving cell boundaries in a fixed coordinate system. If your coordinate system (even without grid deformation), is fixed to a noninertial frame (for example a rotating coordinate system used for turbomachinery), then certainly you have to acount for that by extra terms in the equations. However, this issue is separate from the treatment of grid deformation/motion within your coordinate system. Don't confuse them. 

January 15, 2006, 11:40 
Re: Inertial or Noninertial

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Hi..
Got all your points and thanks "Mani" for ur comment! Let me c if i can put it in order. Lets say that ur solving the flow past an oscillating airfoil(rigid). Lets say u dont wanna deform the grid. One possiblity is we can choose to move the entire grid with the body..of course doesnt seem very attractive so the other possiblity is we go on to the noninertial frame, add the extra terms in the NSeqns that govern the rotation and solve it right away. In this frame of reference the grid is stationary wrt to the airfoil. This is wat is perhaps done for turbine/compressor blade analysis. Now lets say..u wanna deform the grid.."i dont want it to remain stationary" One possibility is sit in the inertial frame, deform the grid according to the airfoil motion, account for the grid motion in the fluxes where the grid velocities are with respect to the inertial frame right ! solve it...The other possibility is sit in the non inertial frame...add the extra terms for rotation etc and incorporate the grid movement in the fluxes wer the grid velocity is wrt to the moving frame !. I hope I got it right.. cheers 

January 15, 2006, 13:43 
Re: Inertial or Noninertial

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Mani wrote: The question is interesting but the discussion it kicked off seems tedious and misleading.
 Hi Mani, I wouldn't consider the preceeding discussion to have been misleading, at all. It was an attempt to find out what terms needed to be added to the NS to compensate for a noninertial reference frame & how to relate motions back to an inertial reference frame  for fluids. I refered to 'rigid bodies' as the simplest form of moving reference frame analysis & as a basis to compare against. Considering B787's additional comments, this would be an appropriate juncture to proved insight on how to perform this. I am also very interested in understanding the approaches used. Cheers, diaw... 

January 26, 2006, 04:12 
Re: Inertial or Noninertial

#16 
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hi this is an interesting discussion....
i just wish to check... for a heaving airfoil e.g. moving up/down with h=h(0)*cos(kt), which h(0),k are just constants, differentiating will give velocity & acceleration. hence, do i need to use a noninertial frame to solve? or can i just move the whole grid in an ALE formuation in an inertial reference frame? regards 

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