Inertial or Non-inertial
 

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 Non-inertial coordinate system or just be in the Inertial coordinate system.

regards,

B787

Re: Inertial or Non-inertial
 

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

Re: Inertial or Non-inertial
 

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

diaw...

Re: Inertial or Non-inertial
 

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

Re: Inertial or Non-inertial
 

Hi Ag,

It is an interesting thread.

How is the moving (translating + rotating) reference frame tied back into the conventional N-S derivations? These are typically derived with a moving inertial reference frame referenced back to a Eulerian inertial reference frame.

In rigid-body 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...

Errata: Inertial or Non-inertial
 

Errata: Should read - for rigid-body 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...

Re: Errata: Inertial or Non-inertial
 

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.

Re: Inertial or Non-inertial
 

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 non-inertial frame would be far more.

Hope this helps

Regards,

Ganesh

Re: Inertial or Non-inertial
 

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 non-inertial system is a shift in perspective, regardless of whether you are talking about rigid bodies or fluids.

Re: Inertial or Non-inertial
 

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 non-deforming 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

Re: Inertial or Non-inertial
 

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 non-inertial frame of reference is quite common (or was) in the turbo-machinery 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).

Re: Inertial or Non-inertial
 

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 N-S eqns basically treat the fluid element a 'flexible-sided' 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 N-S can be derived from first principles in less than 5 minutes.

Remember, even if you have a non-inertial 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...

Re: Inertial or Non-inertial
 

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 non-inertial 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.

Re: Inertial or Non-inertial
 

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 non-inertial frame, add the extra terms in the NS-eqns 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

Re: Inertial or Non-inertial
 

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 N-S to compensate for a non-inertial 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...

Re: Inertial or Non-inertial
 

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 non-inertial frame to solve? or can i just move the whole grid in an ALE formuation in an inertial reference frame?

regards