I agree that the fundamental quantities are based on physical govening laws, based on the Reynolds transport theorem (mass, momentum, total energy) and that other fields are introduced on mathematical basis.

However, concerning enstrophy I don't think its meaning is other than what based on its definition, it is a scalar field that intrinsicly measures the strenght of the vorticity field without a vector implication (that is directionality). Therefore, based only in the scalar value, we can attribute only partially the meaning of intensity of vortical structures as vorticity is possible in a simple strain field with no structures.

More interesting is the discussion about the governing equation of the enstrophy that has several interesting terms.

m2/s2 is equivalent to J/kg which is also the unit of absorption dose, https://en.wikipedia.org/wiki/Absorbed_dose. This may give some clue on the nature of enstrophy.

First, according to Pope enstrophy is defined as the square of the vorticity and the equation for enstrophy is defined by Eq.(2.68). This definition requires to integrate enstrophy over a surface to get the dimension of a specific kinetic energy.

However, as you can see the equation is not derived by an original physical conservation law (such as for mass, momentum and total energy) but it is induced by the Helmholtz equation. It has a the classical convective and diffusive term but further terms appear.

Being enstrophy a scalar field that is related to the intensity of vorticity, you could think that a spectral analysis of the content allow to see the transfer between wavenumbers.

Enstrophy transport equation can be directly obtained from the vorticity transport equation (i.e., from Navier-Stokes equation) and is a measure of rotationality in the flow. We have provided the general derivation of enstrophy transport equation for any flow in:

Diffusion in inhomogeneous flows: Unique equilibrium state in an internal flow - Tapan K. Sengupta, Himanshu Singh, Swagata Bhaumik and Rajarshi Roy Chowdhury, Computers and Fluids, vol. 88, pp 440-451 (2013)

One can derive an instability principle of fluid flow, starting from this equation. That is another work reported in recent time.

However, we often forget that in writing conservation principles of fluid flow, we are happy with translational energy/ momentum. What happens to conservation of angular momentum? Enstrophy transport equation tries to fill up that gap!

Prof. Sengupta,

Thank you for the explanation regarding the enstrophy transport equation. Enstrophy field,, as I understand it, is the surface integral of squared vorticity, hence, its unit dimensions are m^2/s^2. Therefore, it should be a scalar field with no spatial direction indices, is this correct? If yes, then how is it a measure of the flow rotationality? and what is the difference between enstrophy and vorticity in that sense, given the fact that vorticity is a vector field. My question is actually driven by the idea that enstrophy is used to distinguish different regimes of energy cascade (forward/backward or inverse) in homogeneous isotropic turbulence, but I'm not sure how.

Prof. Filippo Maria Denaro

Thanks for the reply. Let's say we have a vorticity field of w(x,y,z,t). To calculate the enstrophy, we compute surface integral of w^s. The result is enstrophy=Ew(t). The space indices is lost in the surface integral, right? So in Fourier space, we would get Ew(f), and we would be required to use Taylor's hypothesis to get Ew(k), only if we have homogeneous isotropic turbulence. My question is: What is the difference between entsrophy in that sense and kinetic energy? they have the same unit dimensions, however, kinetic energy is local in space while enstrophy is not. In short: what is the benefit or advantage that enstrophy could provide to the study of turbulence (especially anisotropic non-homogeneous turbulence).

Dear Khalid M. Saqr

Please take a look at the monograph by Doering and Gibbon on Applied Analysis of Navier-Stokes Equation (CUP, 1995?) where the authors show using periodic flow that enstrophy is related to dissipation. However, our work is for general inhomogeneous flow and not restrictive, and helps explain how coherent structures and patterns form at moderate to high Reynolds numbers due to flow instability.

You also would appreciate the utility of enstrophy over vorticity, when you visualize the vortex street behind a regular bluff body, if you vectorially add the vorticity, you will not get the true picture about the rotationality, as positive and negative component will cancel each other. However, if you look at the enstrophy (square of vorticity), you will see how much of rotationality is created. That is why enstrophy is so important.

Khalid M. Saqr

the definition of the enstrophy scalar field provided by Pope does not contain the surface integral but it is the inner product omega.omega. Even if you perform the area integral, the space variability still holds in 3D. Only in 2D you have the spatial variable saturated by the integral.

Dear Filippo,

Enstrophy is a point property and not an integral quantity, irrespective of dimension. That is how one can write a transport equation for this quantity at any point. To exhibit the physical attribute of this, sometimes enstrophy i integrated over the domain, specially for periodic problem. There should not be any confusion about this.

dear Tapan

unfortunately you find quite common in internet the definition of enstrophy by means the area integral of the squared Vorticity.

I don’t know why, I suppose to make it homogeneous to the dimension of kinetic energy. Or maybe to the use of enstrophy in 2D turbulence.

I agree that is a pointwise property the flow.

Dear All,

Physical sense has only observed quantities, for which in parallel with experimental definition of their values, measurement errors (inaccuracy) can be objectively estimate. From theory of metrology follows that only invariants can be treated as observed physical quantities - for space it is length, for time - interval, mass as measure of a physical object property. For fluids measurable parameters are thermodynamic potentials, values of their derivatives that are density, entropy or temperature, pressure and concentration of dissolved or suspended matter - or partial mass, and momentum as measure of fluid motion. In addition, parameters functionally related with observed quantities are measurable (for example velocity of sound and velocity of light, ...). Velocity of fluid is abstract mathematical quantity which is not conserved in fluid flows (in nature fluid of a homogeneous density does not exist), as well as its derivatives (vorticity, divergence, enstrophy). So enstrophy has no physical sense and must be treated as some mathematical operator for calculations efficiency, as well as circulation.

Professor Yuli D. Chashechkin

Thank you very much for the valuable reply, with which I totally agree.

In turbulence, however, since the physical processes are complex (cascade, dissipation...etc), we expand the range of "physical" variables to include some parameters. In PIV measurements, the velocity field can be reconstructed from accurate measurements of transient particle position. Then, we use mathematical operations to derive numerous flow field variables to represent different physical phenomena. The problem that I currently face is with non-Kolmogorov turbulence. The use of Taylor's hypothesis is impossible. Hence, the energy cascade is only available in frequency domain. Enstrophy appears to be a viable parameter to investigate such cascade in the absence of spatial properties, however, the literature is controversial and scattered. Some researchers use it to inquire energy flux, some others use to to investigate flow "rotationality"... When I asked about the physical meaning of enstrophy, I meant: what is the physical phenomena/process that can best be described by inspecting enstrophy in space/time/frequency domains.

I agree that the fundamental quantities are based on physical govening laws, based on the Reynolds transport theorem (mass, momentum, total energy) and that other fields are introduced on mathematical basis.

However, concerning enstrophy I don't think its meaning is other than what based on its definition, it is a scalar field that intrinsicly measures the strenght of the vorticity field without a vector implication (that is directionality). Therefore, based only in the scalar value, we can attribute only partially the meaning of intensity of vortical structures as vorticity is possible in a simple strain field with no structures.

More interesting is the discussion about the governing equation of the enstrophy that has several interesting terms.

Dear Filippo,

While I agree in principle with your statement that fundamental quantities are based on physical governing laws, one can be little more circumspect about it. What are these physical laws? Aren't these based on conservation principles in fluid mechanics? Unfortunately, in fluid mechanics, due to difficulties in formulation, in energy conservation we only keep translation kinetic energy. To my knowledge, angular momentum may be considered in principle, but a priori we do not have any knowledge about the moment of inertia tensor. Hence we do not have an equation for conservation of angular momentum.

Thus, we introduce the rotatinality via vorticity and one of its measure is enstrophy. There are other measures like palinstrophy. There are elegant theories and descriptions in many books and monographs on such topics. For a beginner, it may be tough to appreciate this, without deeper study of the subject, but ignoring it will be individual's approach to learning! Without vorticity transport equation, it is difficult to explain the role of vortex-stretching as the powerful mechanism for three-dimensionality. In recent times we have introduced the concept of disturbance enstrophy transport equation, that can explain nonlinear instability of incompressible flows in:

Article An Enstrophy-Based Linear and Nonlinear Receptivity Theory

To say that all physical quantity should be measurable is questionable. You never measure entropy, yet it is central to thermodynamics! In fact, we cannot visualize pressure without inserting a probe to measure mechanical pressure. Is mechanical pressure the same as thermodynamic pressure?

These are my two cents worth of understanding enstrophy. All of you have a wonderful 2019!

Dear Tapan

first of all my best wish of happy 2019.

Yes, I see your point but what I addressed before has to do with transport Reynolds theorem. From that, we write the fundamental conservation equations for mass, momentum and total energy. I wonder why do not consider the same approach for the angular momentum. In other words, why we do not write the equation for curl (rho v) from the Reynolds theorem? I can see that the assumption of homogeneous density allows to consider directly curl (v) but what in general? Finally, while the vorticity equation has both vector character and magnitude of vorticity, the enstrophy has only a scalar meaning. However, both vorticity and enstrophy are not necessarily an indication of vortical structure, the classical example of Poiseuille solution has both vorticity and enstrophy. For this reason I wrote that is more interesting to analyse some isolated term of the enstrophy rather than the global magnitude.

Dear Filippo,

Wish you the very best for the New Year! I have thought about the issues raised in your reply. One answer that we never can state conservation of angular momentum is related to inertial property: For translational energy, the inertial property is mass and which is invriant! It is not so with the angular momentum, as I explained in my last post about our inability to obtain the moment of inertia as function of time. This is also part of the answer we are looking for!!

Finally, enstrophy is a point property, so there may be some special cases for its integral. But that is not important, as we have also cleanly written down the enstrophy transport equation in:

Article "Diffusion in inhomogeneous flows: Unique equilibrium state ...

Please read it and give your critique!

Thus, we cannot write a conservation equation of the type

d/dt Int [V] rho curl v dV = Int[S] ... dS

But conservation should apply in a closed system where no torque is applied, isn't it? Interestingly, this physical properties are quite generalized also for fluids in Wikipedia (https://en.wikipedia.org/wiki/Angular_momentum#Conservation_of_angular_momentum)

What is quite strange is that we use only mass, momentum and total energy as a closed system of equations. The key to add to the system the angular equation is to have this additional unknown inserted in a different closure for the newtonian model of the stress, right?

Dear colleagues, It is my time to wish everybody a Happy and Constructive new 2019 Year!

The problem of angular momentum definition is related with mobility of atoms, molecules, their aggregations and clusters in fluid media. As result any flow is characterized by intrinsic internal structure and spatial scales of the structure components are defined by dissipative coefficients and characteristic time of formation or period of the process. As density of internal energy is increasing with lengthy scale decreases, filament structures can be visualized in any flow.

"Concerning In PIV measurements, the velocity field can be reconstructed from accurate measurements of transient particle position" - it is so only for particles velocity. Another particles will give give distinguished flow pattern. Uncertainty of results increases with decreases the particle size.

In general case velocity of a fluid and all its derivatives is universal mathematical symbols but cannot be observed experimentally, as well as angular momentum.

Filippo: That is an interesting thought that angular momentum conservation can be embedded in stress-strain constitutive relation! Do you have any clue, how one should go about it? I am perplexed, truly.

Yuli: Introducing angular momentum via internal energy is equally attractive. But are there definitive works or references to that? I am not too distracted about limitations/ uncertainty of experiments. So it does not really matter if experimentalists do not measure some quantities at all. What it says though is about great scopes to develop that branch of investigations!

When I was a student I learned a synthetic lesson about the angular moment: do not consider the angular equation but assume symmetry in the stress tensor. From that time I was wondering what if one relaxes this constraint and in some way couple the angular momentum equation written from the Reynolds theorem...

I never explored such idea and I cannot say if it somehow valid or not...

Dear Filippo:

We will be eager to know the results of your explorations......

Sounds like you have some clue after all! Look forward to hear more from you on this.

At present it has only the appearence of the idea in the anegree of Plato's cave ... :-)

However, I would think considering the relation between stress tensor and gradient of velocity while adding a further term coupled to the angular moment. This way the result that the stress tensor is or is not symmetric would depend on solution of the angular moment and is not a general constraint... Actually, I think that if one explores the literature in the physics field some scientist could have already explored new ways....