### Brief Note on the Development of the Framework

Fluid dynamics often relies on the Navier-Stokes equations (NSE) to model various flow phenomena. I have here a theoretical framework which recasts the equations of fluid mechanics in a Lagrangian formalism in a way where it can be shown that the kinetic energy of the vorticity explicitly break the scaling symmetry of the theory. This explicit symmetry breaking potentially provides a mechanism by which energy cascades and dissipation occur in turbulent flows. Additionally it demonstrates how the inclusion of vorticity fixes a scale.

This framework was developed from abstract symmetry principles and aims to address multiple problems in physics. My first goal was to understand how motion couples to the underlying geometry resulting in emergent dynamics. Fluid dynamics are fundamentally “shaped” by geometry, so I began by exploring simple fluid systems.

Some brief calculations already suggest the role of local scaling symmetries. If we define a conservative vector field to model a wave propagating in a pipe with cross sectional area

we know that it is by definition irrotational: . The mass flow field is also irrotational for constant cross sectional areas . Thus the field exhibits self-similarity at a scale (for any choice of ). However if we locally fix a scale by promoting to a varying cross sectional area, we see that the mass-flow becomes rotational This is a very basic derivation but I hope the results are clear, locally fixing a scale introduces additional dynamics, particularly rotational dynamics.

In a similar manner, by demanding that a real scalar field be invariant under local scaling transformations a rotational vector potential must be introduced resulting in a Helmholtz-decomposition. An interesting consequence is that the symmetry of the Lagrangian is explicitly broken by the kinetic term for the vector potential, again suggesting the association between vorticity and scale-fixing. Another result is that the homogeneous equations for the vector potential imply that the field is irrotational and thus conservative.

### Theoretical Background

#### Scaling Symmetry and CFT

Delving deeper into the symmetry where is the scaling factor, ties to conformal field theory (CFT) are considered. Global scaling symmetry is almost universal as it corresponds to the trivial case where is a constant, the more relevant and interesting cases are the local scaling symmetry that occurs when becomes a function of spacetime coordinates .

We recognize that the scaling transformation is a conformal dilation, indicating that the group of scalar matrices is a subset of the conformal group : This conformal dilation is a Weyl transformation, defined as the local scaling of the metric tensor: where the conformal factor generates a new metric in the same conformal class as . The field has a conformal weight , where . Identifying with , we can conclude that . Combining Equation (1) with

we have two transformation laws. In order to understand how transform, we need to formally define the theory.

### Formulation of Scalar Field Theory

To ensure that the symmetry is respected, the covariant derivative is introduced, where the field has components of pressure and velocity. In order for the derivative of to respect the symmetry we need to transform the same as so or equivalently . To satisfy Equation (3), we conclude that must transform as where . Now we have the full set of transformations, Equations (1), (2), (3) and (4)

We can then modify our Lagrangian for a free scalar field so the minimal coupling for this theory is Equation (5) has the equations of motion (EOM) where . Expanding the terms we get

Note that both Equations (5) and (6) obey the conformal symmetry we have outlined.

### Explicit Symmetry Breaking

Equation (5) describes the minimal coupling but for a complete description of the system we need a kinetic term for . I defined the term where as is typical for vector fields in gauge theory. Note that is the vorticity tensor, and that the term is not symmetric under the conformal transformations

This means that including the kinetic term for the field results in the explicit breaking of the conformal symmetry and thus fixes a scale. The total Lagrangian

can now be used to describe the whole system. The EOM when varying with respect to are which can be combined with Equation (6) to obtain

Here Equation (6) respects the symmetry whereas Equation (9) does not suggesting that the scalar potential field maintains its invariance.

Some calculations reveal that the current can be defined as
where
is the canonical momentum. Expanding

it becomes clear that
couples quadratically to the rotational field. Further
investigations suggests the current to be associated with mass-flow
rates, but first we need to use the framework outlined to actually
represent fluid systems before we can further analyze.

### Fluid Representation

It may not seem immediately obvious that Equation (8) can represent a fluid system, indeed it took me a while to come to that conclusion. Especially in the case of the field which I struggled to interpret physically for quite some time. I have landed on a Maxwell-like representation that I am satisfied with, and so far the results have been promising. Indeed the theory already greatly resembles classical electrodynamics so why not continue with the analogy?

The scalar potential is easy to represent. The state of a fluid can be given by it’s pressure and velocity and respectively so we define the four-vector which has components where is the speed of sound. For the time being we will work dimensionless to simplify the calculations, the constants will be relevant later. Combined with the incompressible field equations derived from the NSE we have thus adequately describes incompressible flow consistent with then NSE. Note that the EOM of this theory, , are that of a free scalar field which will become relevant later for gauge fixings.

For the field I used a Maxwell-like approach by defining two vector fields in 3 dimensional space and which I have denoted the transport (or convective acceleration) and vorticity respectively. Similar to electric and magnetic fields

Where is analogous to the electric field and the magnetic field. These fields allow us to represent and analyze rotational flow in ways that are somewhat familiar. Using Equations (13), and (14) we can reinterpret Equation (9) as a set of Maxwell-like equations and their integral forms

Here the intricate interactions contained in Equation (9) can be appreciated with this representation, where they manifest as exchanges between and .

Additionally it follows from Maxwell theory that there exists stable self-propagating waves which I am, creatively, calling UW waves. Taking the curl of Equations (17) and (18) we get

I think that UW waves could be useful for describing large stable structures in turbulent flows, however more validation is needed. Vortex shedding is a particular physical case I can think of where there are stable self-propagating vorticity waves . The shedding of vortices in alternating directions reflects the Maxwell-like nature of this system. It is also relevant that rotation in the system explicitly breaks the symmetry of Equation (8) which fixes a scale. This is particularly relevant as it aligns with known concepts in the study of turbulent flows.

## Comments

This is just a sort of general overview, I will gradually edit and add to this post.

After a discussion with Eowyn, this is a correction to my post on Codidact.

Again, I want en equation of the uniform flow. Again vector potential is zero. And again we are left with:

This is the place where I was wrong in my Codidact’s post. I did not pay attention to the fact that we are using tensor notation here and just took corresponding derivatives. That was a mistake. Eowyn educated me on what this term really means.

It turns out, this is d’Alembertian. Eowyn showed me a slightly modified version of d’Alambertian which does not include the speed of light . I guess it means he is working in the coordinates where . In turn, that means that we have to be careful trying to get the governing equation of the flow.

Let me just copy the derivation from Wikipedia where no assumption of coordinates is made, but use Eowyn’s sign convention for metric :

Please correct me if I am wrong, but to the best of my limited knowledge of GR, if we neglect effects of relativity (for the case of regular flow of liquid that we observe on Earth in our every day life), the last expression reduces to:

The right hand side is, of course, a Laplacian:

Substitute Eqn. for the right hand side of Eqn. and get:

Now, plug in Eqn. into Eqn. and get the equation of motion:

From Helmholtz decomposition, it follows that and Eqn. becomes:

Note that Eqn. is true for any irrotational flow. Let’s simplify it for our case of classic uniform flow. Classic uniform flow is a parallel flow along the direction. Therefore, Eqn. can be expanded and reduced as:

Integrating Eqn., we arrive at the equation of velocity field:

Which is, indeed, the equation of the classic uniform flow.

Couette flowIn my previous comment I talked about Uniform flow. The next baby step should be Couette flow, I think. That is because Couette flow is very simple, yet rotational allowing us to clearly see how exactly vector potential works out in Eowyn’s frame work.

Let’s consider the absolute classics: flow between two horizontal plates. The bottom plate is fixed. The top plate moves with velocity to the right and drag the liquid creating the flow. The distance between the plates is . is to the right, is upwards, the origin is placed at the bottom plate. Assume steady state flow, only component of velocity (i.e., there is only projection of the velocity vector), the flow is uniform in the direction (i.e., no velocity gradients in the x direction).

A quick reminder of how we normally solve it.Start with the projection of the N-S (i.e., the Navier-Stokes equation) on the axis:

Due to our assumptions, and we are left with:

Integrating it twice with the parameters I gave in the assumptions, we get:

And the reason Couette flow is rotational is because

Note, that Couette flow has only one vorticity component directed perpendicular to the flow plane. I will make use of it later.

Now let’s see if we can get the same solution using Eowyn’s framework.Again, we start with Helmholtz decomposition: . Here, is the velocity field, is the flow potential, is the vector potential, is the rotor operator (sometimes refered to as curl operator ).

And again we keep in mind that Helmholtz decomposition does not exist for any vector field and it is not unique.

Here - as opposed to Uniform flow - we cannot neglect because we know beforehand the flow is rotational (or at least, we do not have any grounds to make an assumption of the irrotational flow).

Let’s simply first Eowyn’s equation of motion (see the first equation in the system of equations shown between equations (9) and (10) in Eowyn’s original post, as well as Eqn.(6); also see my comment on Uniform flow).

The last equality in Eqn. came from the vector identity

Under the made assumptions, the parameters in Eqn. are reduced as following:

The last expression is tricky: the question is which component of we must keep. From the Helmholtz decomposition given above, one can conclude that must be perpendicular to . Since we have only component of velocity, that reduces to . Evidently if had only component it would not be perpendicular to . So we are left with the choice of either or . Our flow must stay within the plane. Therefore in order to meet the right hand rule, must be directed in direction. Therefore, must have only component.

With that in mind and remembering the identities of Helmholtz decomposition, Eqn. reduces to:

The reason was assumed in Eqn. is due to the assumption of flow uniformity in the direction . According to that, since , the gradient of must also be invarient of .

Now let’s simplify Eowyn’s second equation of motion (see the second equation in the system of equations shown between equations (9) and (10) in Eowyn’s original post, as well as the equation right after Eqn.(10)).

Honestly speaking, it took me several attempts before I managed to rewrite Eowyn’s second equation of motion in the matrix form correctly (I hope I got the right version...).

Vorticity is given by the general equation . Plug it into the equation Eqn.: