But in both cases the air moves relative to the airfoil. If you watch what happens from the airplane, the airfoil is at rest relative to you, and the air comes at you, just as in the wind tunnel. If you watch from the ground, the airfoil comes at the air. Galilei invariance makes sure that the predictions (the description through equations) for what happens are independent from the observer frame (from the ground, from inside the airplane; from the the wind tunnel, or from the air moving through the wind tunnel). Boltzmann equation, Navier Stokes equations, and any other transport equations are Galilei invariant. Indeed, Galilei invariance is one of the building blocks used in theories of continuum mechanics when it comes to constructing meaningful constitutive equations for, e.g., stress and heat flux.

Dear Henning Struchtrup

thanks for the reply, what I am highlighting is that if you have a mass of fluid moving in a wind tunnel it has an amount of kinetic energy that encouters an aifoil at the rest. At the wall, the friction acts to dissipate such kinetic energy at small lenght scale. Actually, a real airfoil has its kinetic energy and encounters the fluid at the rest (okay, it has a own movement due to the proper turbulence in atmosphere, but let we disregard) and the friction acts to accelerate the fluids. While the appearence of Galilean invariance seems clear by a macroscopic point of view, is that really correct by a microscopic point of view? Is the mechanism of energy transfer (kinetic to internal energy) well represented? When the Mach number is quite high (hypersonic flows), such discrepancy becomes clear also by a macroscopic point of view.

Thanks for yout attention

Filippo

Hi Filippo

But that's just the thing with Galilei invariance: kinetic energy and momentum depend on the observer (e.g., moving with the airfoil, or the air, or arbitrarily, in an inertial system). Macroscopic or microscopic point of view makes no difference, all equations of motion and energy are Galilei invariant--this is why I mentioned Boltzmann equation (microscopic) and Navier-Stokes (macroscopic).

But maybe I don't understand the question--so, to clarify, let me ask this: Do you see a difference between the airfoil in the wind tunnel, observed from the rest frame of the airfoil (laboratory frame) and the airfoil in flight, observed from the rest frame of the airfoil (i.e., from the airplane)?

Best,

Henning

Hi Henning,

my point is that if you see the kinematic by alone, no matter about the Galilean invariance. But if you have a mass of fluid moving, it has an energy content that is exchanged with a body. Obviously, a real airfoil has its kinetic energy that transfer to a mass of fluid. At the level of the friction, is this difference relevant in the energy exchange?

This fact is immediately clear when you compare wind tunnel experiments in hypersonic flows and real hypersonic vehicles. For example, the plasma wind tunnel generates a mass of fluid that impacts on the body, this flow being, however, already at high energy level, dissociated and ionized. Conversely, a real vehicle impacts a fluid in equilibrium and at the rest, the high temperature with dissociation and ionization happening only behind the shock. No Galilean invariance at all in the energy and thermodynamics processes.

Hence my question, only when we arrive at high Mach flows the lack of Galilean invariance is relevant?

Hi Filippo

I'd say what you describe is not a question of Galilei invariance but of different physical state of the gas that impacts the body. Then, of course you have different interaction between gas and body. The aim of a wind tunnel should be to create the same conditions of the incoming flow as what is present under real conditions--if that's not possible, there is a problem, of course.

Galilei invariance only speaks about transformation of observational frame, nothing else. All equations you will use to describe your plasma impacting the body in the wind tunnel, or your air flow hitting a supersonic aircraft and being dissociated etc in the shock are Galilei invariant. Comparison is limited, however, if the incoming flows have different state.

H.

And, without going up to the hypersonic case, don't you think that a mass of fluid having a certain amount of kinetic energy to trasfer to a solid body is at a different state from the same mass of fluid at the rest? When we study transition to turbulence, when we try to model the dissipative effects of the unresolved scales assuming energy equilibrium, has this difference of state some relevance ?

There is not per se a state "at rest". Motion is relative (to the airplane, to the air, to the wind tunnel, etc.) Do you know the feeling when you sit in a train at a station, and there is a train on the other track, and one of the trains starts to move, and you are not sure which one (until acceleration is sufficiently high to notice)?

I just talked to a colleague from fluid mechanics, and confirmed that in a wind tunnel one of course tries to mimic in the incoming flow of air the conditions that one finds under operation of the air craft (say). Now, if your preparation of the incoming air is "not good" in some sense, then differences will occur--but this is not a question of Galilei invariance, but of different state of the incoming air (e.g. turbulence etc). Also the air up high where you fly will not be at rest, but might have all kinds of deviations from a proper rest state.

Different states certainly will lead to differences in experiments, but it's not a question of Galilei invariance, but rather of the specific states. These make a difference, of course.

Ok, thanks, maybe the fact I denoted this question under "Galilean invariance" is misleading. What you stated is indeed a problem in matching the BCs. With my question I was wondering if there is something hidden that can imply a limit in the assumption of the linear Newton law that defines the link between stress and velocity gradient. Of course that would imply also consequence on the eddy viscosity closure model.

From wikipedia:

Work, kinetic energy, and momentum

Because the distance covered while applying a force to an object depends on the inertial frame of reference, so does the work done. Due to Newton's law of reciprocal actions there is a reaction force; it does work depending on the inertial frame of reference in an opposite way. The total work done is independent of the inertial frame of reference.

Correspondingly the kinetic energy of an object, and even the change in this energy due to a change in velocity, depends on the inertial frame of reference. The total kinetic energy of an isolated system also depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass. Due to the conservation of momentum the latter does not change with time, so changes with time of the total kinetic energy do not depend on the inertial frame of reference.

By contrast, while the momentum of an object also depends on the inertial frame of reference, its change due to a change in velocity does not.

Seems we converge now. Note that the link between stress and velocity gradient is Galilei invariant: All observers in inertial systems (no acceleration) agree on the velocity gradient (apart from, possible, a rotation if they use different Euclidian bases).

Regards to Napoli, and Ciao, H.

I'm by no means an expert yet here is my understanding. The Galilean invariance principle relies on two ingredients: Definition of the velocity and invariance of physical laws due to a coordinate transform. Note that the latter is in fact equivalent to conservation of momentum (Noether's theorem). Now assuming that time is independent of the frame of reference (i.e. non-relativistic approach), combining the two mentioned points give us the Galilean invariance principle. Therefore as long as we have momentum conservation and non-relativistic speeds, we should be able to use the Galilean invariance principle. In other words if we come up with a non-relativistic fluid model which is not Galilean invariant, then I guess we would be violating momentum conservation.

And what about the presence of the acceleration in a studied problem? From wikipedia you can read a statement about “isolated system”. What does that imply?

I think as long as we can cast the evolution of the constituent particles in a Hamiltonian form, we should be able to apply the Galilean invariance principle. This definitely depends on the origin of the acceleration. For example if the acceleration is path dependent, then probably we cant use the princinple.

And what if the acceleration is responsible of a situation in which turbulence is not in energy equlibrium ? I have a lot of doubts about some particular situations wherein there is a non vanishing time derivative of the kinetic energy...

Galilean invariance is built-in in all equations of classical mechanics, regardless of whether they relate to particles, solids or fluids. In other words, the equations of classical mechanics are invariant under transformations of the Galileo Group: translations, rotations, time-shifts, inversion of coordinates, inversion of time, and (galilean) composition of velocities. Only a velocity high enough, compared to the speed of light c, destroys this invariance.

However, the Maxwell equations are not invariant under the Galileo Group. They are naturally invariant under the relativistic Lorentz Group (or Poincare if you take into account the space-time displacements). Therefore, a physical situation involving plasmas, that is, electromagnetic fields, would not be invariant under galilean transformations. Perhaps this is the cause of the discrepancy you see when changing the reference frame in a plasma experiment.

Dear Javier,

Thanks for your contribution. As I asked before, if we have an accelaration we take that into account, as it happens for the NSE with the Coriolis acceleration we explicitly consider itno the equation. But has that some consequences also on the modelling of turbulence when a statistically energy equilibrium is not present?

Dear Filippo Maria Denaro, could year with me more info about the galilean invariance? Im very interested in this topic, but i dont the knowledge to discussed. I will appreciate if you can share with me more info about his phenomena.

This principle is fundamental in classical mechanics:

https://en.wikipedia.org/wiki/Galilean_invariance

Dear Filippo,

I am not sure to understand your arguments. In the end, modelling turbulence is equivalent to modelling the Reynolds stress as a function of the mean velocity. If your choice of reference system affects the definition of u'_i, then is affected, and your modelling must have this into account.

Anyway, I am not sure to understand what "turbulence is not in energy equilibrium" means. After all, turbulence is chaos, the opposite to equilibrium. Perhaps your concern is about simulating an unsteady flow. If your question is: "can I simulate an unsteady flow where acceleration is present?", the answer is "yes, you can, as long as you are careful enough". I have done it myself.

I don't know if this helps you at all, since I am not aware of the details of your problem.

Turbulence is considered in energy equilibrium at the lenght scales below the cut-off of a LES filter, that means thae energy production and energy dissipation balance each other in statistical sense, without a time evolution of the energy. This assumption is quite strong but if we remove that we introduce the doubt I addressed about the acceleration.

If we look at the Wikipedia article on Reynolds stress, we see that the velocity u is split into an average ubar and a deviation u', such that average of u' vanishes. With this, u' is a difference of velocities at the same location (u'= u - ubar) and as such is Galilei invariant. The easiest interpretation is that u' is the fluctuation measured from the particular observer that moves with ubar. I'd guess that all quantities relevant in turbulence, in particular the energy stored in turbulent motion, are measured in the co-moving frame, hence are invariant. This is similar to peculiar velocity in kinetic theory. Again, Galilei invariance ensures that all observers agree on the same outcome based on the same equations, although they move against each other.

If I'm not wrong, I guess we all agree on the fact that the governing fundamental laws of fluid dynamics are Galilean invariant. Now a possible question is: why an up-scaling of those laws should still remain Galilean invariant. Well I guess it depends on your up-scaling operator. As Henning mentioned, Reynolds averaging up-scaling preserves the Galilean invariance property. However I can imagine that one can come up e.g. with a spatial filtering which may not preserve the spatial symmetry (e.g. due to an anisotropic cut-off) and hence violates the Galilean invariance property. If it would be a relevant model or not, it's quite a different discussion.

Following your useful contributions, I would address that in turbulence other decompositions are used, not only the Reynolds one U(x) + u'(x,t). For example, in LES the decomposition implies a local filtered velocity so that one defines u_bar(x,t) + u'(x,t). Furthermore, talking about the kinetic energy you can see we can define it in several ways according to these variables. For example the total energy E=1/2 rho |v|^2 + e is filtered and defines

E_bar=1/2 (rho |v|^2)_bar + e_bar

where (rho |v|^2)_bar is not (rho |u_bar|^2).

I wonder also if an irreversible process implied by friction that transform kinetic energy in internal energy (and increases entropy) has some dependence on the invariance. This process is what we model generally by the eddy viscosity model.

Just as a further contribution to the discussion, this paper analyses the relevance of Galileian invariance under discretization of NSE.

https://home.aero.polimi.it/quadrio/papers/2013-JCP.pdf

what do you think about?

Dear Filippo,

Now I see your point. During acceleration the turbulence intensity diminishes; this is called relaminarisation by some authors. Thus, a measurable decrease of turbulence intensity implies a corresponding decrease in turbulence energy production. At least this result you can take for granted. The question is, would the turbulence energy dissipation also diminish, and at the same rate? Should this happen, your assumption of "energy equilibrium" would still be valid. I have a feeling that the energy equilibrium would be maintained, but I cannot be sure.

However, this has nothing to do with Galilean invariance. The relaminarisation is a phenomenon related to an increase in the mean velocity , U_i, without a corresponding increase in the fluctuating velocities, u'_i, resulting in a decrease of the turbulence intensity, u'/U. It is a dynamical effect of its own, not related with inertial reference frames .

Dear Javier

yes, apparently Galilean invariance is not involved, you are right. But, again, think about in terms of the thermodynamic system, specifically considering the entropy variable. Can we think that a moving mass of fluid having a certain amount of kinetic energy that is transformed in a part of internal energy when encounters a body has the same energy process of the fluid at the rest with the same body in movement? And what when we consider the various form of filtered kinetic energies?

The paper I posted analysed the effect of the discretization. However, I would think that there is something more "sneaky" than simply the non-Galilean invariance in the discretization..

In my opinion the angle is influenced by reflections that can multiply the light, the reflection is caused by concave and convex light, as well as from clear water that can react to any reflection anywhere.

Hi Filippo,

I may be late to the party and completely off-base, but I had a thought that may help. Javier certainly seems to have the answer in hand, but I thought I'd add to it.

If I understand the question correctly, you are wondering if there will be a difference in the energy transferred from your fluid to your airfoil between two cases: 1) Foil at rest, wind going past and 2) Wind off, foil moving past. As you have pointed out kinetic energy of the air in these two cases is different, implying that the energy change could be different. Kinetic energy is inertial frame dependent. Like potential energy, kinetic energy is not an absolute, and only affects the world in ways that rely on how it changes, not the number assigned to it. The "physics" (change in internal energy) should be the same whichever interpretation you use, since it should be a function of the difference in velocity between air and airfoil. This difference should not be reference frame dependent.

Put another way, your two situations appear to be two physically different cases (one where the foil moves, and one where it's still), but I think they are the same physical case. 1) the observer is fixed in place as your wind accelerates past a stationary airfoil or 2) The observer moves downstream at some acceleration to make the air appear to have no "macroscopic velocity". As far as the equations and your data will be concerned this is the exact same situation as you've described, but I have not changed anything that is physically happening, only changing how I interpret what happens.

Now in case you are considering an LES simulation specifically. I suppose it could matter which interpretation you use. In particular I would wonder if the two situations would be different in terms of how much of the internal energy transfer would be greater than filter scale (macroscopic), and how much would be less than that scale (microscopic). My intuition is that could be dependent on reference frame, but it should still not change the physics, since the dichotomy between micro- and macro-scopic is also artificial.

Hi Thomas

many thanks for your contribution. I see your point and this is the classical framework I always accepted, to. My question is due to a twofold thought: a) the possible difference in the energy process in terms of entropy production in both cases and b) the fact that Galilean invariance is sometimes not fulfilled in LES codes but the results seems to not be sensible to that.

Hi Filippo,

Okay, so long as that much is clear. I could certainly see that LES codes may not satisfy Galilean invariance. If for no other reason than the notion that "what's macroscopic and what's microscopic" is almost certainly dependent on which reference frame you are working from. Still, if the small scale modeling is good it should still result in a pretty similar answer in both situations. I'm a bit too rusty on entropy to be able to comment on anything more.

Thomas, the paper on JCP I posted above analyses the difference existing in performing the DNS using the moving (convecting) box and the fixed box. Therefore we can assume that is not only the SGS model to act on. But I still have some doubts about why a moving reference should increase the quality of the results.

For this reason I started to think to a more general and physical comparison between the cases of an airfoil moving in air that is at the rest and one fixed and impacting a flow in a wind tunnel. I see that all of us would accept the assumption of Galilean invariance without too much care, but when we transfer kinetic into internal energy at the cost of an increasing entropy, is this mechanism to be modelled totally equivalent in our numerical models.

Again, maybe mine is a stupid doubt ;-)

The algorithms that LES uses to generate the subgrid scales, with cut-off operations, and subgrid-scale turbulent viscosity assumptions, are actually very artificial and detached from physical principles. Therefore, I would not be surprised to learn that they do not satisfy the laws we take for granted, be it Galilean invariance, or "energy equilibrium", or even the Second Principle of Thermodynamics. We can be sure the underlying Physics MUST fulfil the relevant laws and principles, whereas the numerical procedures can fail to meet some of them, due to the artificiality of their inception. I am afraid the only way to know what will happen to the numeric schemes is to actually calculate two cases differing exclusively in the aspect your are in doubt with, and compare the results. Even then, you cannot be sure the results are universally applicable, but probably will be case-dependent.

it seems logical at first that GI holds, however, as we know from the discussion of time directionality and thus indeed the probability of states - or in short the discussion about entropy and syntropy I would predict that as usual in might hold in first approximation but not while looking at it closer...

...I think also the system critically matters here: big scale? or molecular state? in the later I am quiet sure, GI is violated, with all its consequences...

very good ,I THINK SOO,

Dear Prof. Filippo Maria Denaro, in the development of the microscopic theory of superfluidity, that is fluid motion without viscosity, the Galilean invariance assumption allowed L. Landau to carry out successfully a mathematical theory of superfluidity that explains the properties of liquid helium at a temperature below 2.17 K.

The theory of L. Landau is based on the Galilean invariance of energy transformation and impulse. Using the Galilean invariance, he established the criterion for superfluidity, that is, the relative speed between fluid and capillary is smaller than the critical value, V < Vc . Above Vc viscosity or dissipation exists in 4He.