Could anyone expalain What is Galliean invariant? Thanks.

Bill

Hi Bill,

I am not sure if this is completely related to your question, but in physics: if linear momentum, angular momentum and energy are conserved in one Galilean frame of Reference, then these quantities are conserved in all such framces. Now a Galilean frame of reference is a frame of reference that is not subject to any acceleration or rotation, but it might be in translation at a constant speed. For example if you are on a train, and the train is moving at a constant speed and on a straight line, then the train is a Galilean frame of reference. If momentum and energy are conserved for a systme on that train, then they will be conserved also on any other such Galilean frames of reference (it could also be at rest at the train station, in this case the speed is just = 0). So I would expect that the Galilean invariants are the energy and momentum. But I might be wrong, since I never heard of that in CFD.

I hope this helps. Any body else has something to add about that?

Cheers, Patrick

Actually, one can derive the linear momentum equation by requiring Galilean invariance on the energy equation. To demonstrate Galilean invariance, one must show that the form of the equation is unchanged in respective reference frames which are related by a Galilean transformation (i.e. moving at a constant relative velocity with respect to each other). In a general sense this is done by considering a change of variables

x* = x + Vt

u* = u + V

t* = t

where V is the relative velocity. From a physical point of view Galilean invariance simply says that the form of Newton's laws of motion would be the same in any reference system that is non-rotating and non-accelerating. You should also be aware that not all things are invariant under a Galilean transformation - relativistic quantities satisfy a different invariance principle.

>> since I never heard of that in CFD.

It is invoked quite a lot in fluid mechanics when discussing the forms of modelled terms. For example, it pops up quite a lot in the area of turbulence modelling.

A quick google suggests just under 50% of the hits (turbulence OR fluid OR voriticity) are to do with fluids rather than mechanics/physics. I find this a bit surprising but perhaps there is just more volume in the area of fluids?

Patrick,

Momentum is not Galilean invariant. If a body is moving at a constant velocity in a straight line, then the momentum of the body as seen by some one stationary would be different from the momentum as observed by some one moving along side the body.

Conservation of linear and angular momenta have to do with the fact that equations of motion are invariant under arbitrary linear and rotational transformations. The new coordinate system is stationary (and not moving). eg. (X,Y) = (x,y) + (a,b) is a linear transformation where a,b are constants. Energy on the other hand is a scalar and should not depend on the coordinate system (as long as it is not moving).

Galilean invariance is a stronger condition. This says that some quantities are invariant under linear translational motion of the coordinate system. eg. (X,Y) = (x,y) + (a*time,b*time). A stronger condition would be material frame indifference which means invariance under all types of rigid body motion of the reference frame (arbitrary combinations of linear translational and rotational motions).

I am guessing that the original question was asked in the context of subgrid/Reynolds stresses in turbulence. They are material frame indifferent since velocity gradient tensor can be decomposed into two components : a strain component and a rigid motion component and the latter does not contribute to production of turbulence (i.e. energy transfer to small scales).

You missunderstood me,

I said that if the momemtum is conserved in a Galilean system, then it will be conserved in any other Galilean system, but I did not say that the momemtum will have the same value in all the different Galilean systems....

Thanks a lot to Patrick, Greg, Andy, and Kalyan. I have benefited from your talking.

Bill

My oringinal question is what is the definetion of Galilean invariants. In fluid mechanics, which belongs to the Galilean invariants in the following parameters: velocity (momentum), kinematic energy, potential energy, thermal energy(temprature), stress, and velocity gradient.

According to Greg and Kalyan explained above, Galilean invariants are those parameters which should be conserved to be constant(for a given position in the frame of reference) from a Galilean system to another Galilean system.

Thus,only thermal energy(temprature), stress, and velocity gradient are Galilean invariants.

Velocity (momentum), kinematic energy, potential energy are not Galilean invariants, because they will change the values in different Galilean frame of references.

Is this right? Can anybody make more explanations? Thank you.

Bill

Maybe I am wrong, but I have never seen anything concerning Galilean invariant, but a lot concerning Galilean invariance.

The question is more "Does my set of equations is the same when I look to the decay of the turbulence intensity in my cup of tea in the train (linear - constant speed) or in my living room ?"

Of course, a cup of tea in the train has a greater global kinetic energy than the cup of tea in my living room if I look at it from my living room. But in term of turbulence the two cups share the same behavior.

Then turbulence quantities look as if they are Galilean invariant - following your definition. This is not surprising since they are defined using the Reynolds decomposition which consist in substracting the mean value of each quantities.

Just to say someting...

Sylvain

PS : I don't have any reference about it, but if you are dealing with turbulence and Galilean invariance, you may take look at paper from Shih Zu and Lumley for a part and from Speziale for a another part in the early 90's.

While it is possible to consider various quantities as being invariant, invariance principles are more useful when applied on a broader scale, as Sylvain points out. The notion that the equation for the conservation of energy must be invariant under a Galilean tranformation means that two researchers operating independently in coordinate systems moving relative to each other with a uniform velocity would "discover" the same conservation law, namely energy is conserved. Because of these invariance principles, we can be assured that the conservation laws that we use don't have to be rediscovered everytime we change our point of view.