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In turbulence, the enstrophy of a flow in a domain $\mathcal{D} \subset \mathbb{R}^{D}$

$$ \mathcal{E} = \int_{\mathcal{D}} |\vec{\nabla} \times \, \vec{v}|^2 d^{D}x $$

appears sometimes, it's cool and has cute properties in 2d (see this answer about turbulent flows), etc.

Now, what does it represent physically? Okay, it's the integral of the vorticity squared, but has it a physically understandable interpretation? Why is it interesting to study that?

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I mean, if it has a name, it should have a meaning. – Georg Sievelson Feb 11 '13 at 20:13
@Emilio are you sure of your edit ? For a 2d flow, for example, the (total) enstrophy is a surface integral over the whole domain, not a line integral (and for a $D$-dimensional flow, it is $\int_{\mathcal{D} \subset \mathbb{R}^D} |\vec{\omega}|^2 \text{d}^{D} x$), with $\vec{\omega}$ the vorticity). (We could also consider the local enstrophy $|\vec{\omega}|^2$ ...) – Georg Sievelson Feb 12 '13 at 18:14
No, I am not (though the wikipedia article does indicate that). I was just put off by an integral without a domain, measure, or dimensionality. – Emilio Pisanty Feb 13 '13 at 11:02
Well, it was supposed to be an integral on the whole domain. I will correct it. – Georg Sievelson Feb 14 '13 at 19:18
Comment to the question (v4): Please clarify the appropriate definition of the cross-product $\vec{\nabla} \times \, \vec{v}$ if the dimension $D>3$ is bigger than three. – Qmechanic Jul 23 '14 at 10:21

2 Answers 2

From mathematics point of view, it is a surface integral of the scalar quantity |Curl v $|^2$. The physical meaning of it, in the context of fluid dynamics in 2-D or 3-D, is that it has the units $(m/s)^2$ which when multiplied by the density of a fluid represents some form of energy. As for the meaning of the intgral in a D-dimentional space, the problem is more of mathematics than physics. For a physical interpretation of this integral in D-dimensions, one needs to know first what the Curl v itself could represent, in a similar context to that of fluid flow in D-dimensional space, and then wonder what the physical meaning of its intgral on D-dimensional surface could mean. Its dimensionality, $(m/s)^D$, does not seem to relate to something physically meaningful for arbitrary values of D. I am not aware of a good physical analogy of this in D dimensions.

As for the the word itself, it derives from the Greek $\epsilon \nu - \sigma \tau\rho o\phi\eta$. It means: "during the rotation",in a state of rotation or "while turning". In other words, the word makes reference to turning, rotation. I hope this helps you visualise the meaning of the word in relation to the turbulant motion of a fluid.

Similar to the way Entropy is derived: $E\nu\tau\rho o\pi\eta$ which means: in a state of change, during a change.

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I'm not sure to get your point ... if I consider a $D$-dimensional flow, $\mathcal{E} = \int |\vec{\nabla} \times \, \vec{v}|^2 \text{d}^{D}x$, and then the dimension of $\mathcal{E}$ is $[\mathcal{E}]=L^{D}T^{-2}$. For e.g. $D=3$, which is quite acceptable, $\rho \mathcal{E}$ is not an energy nor an energy density (with $\rho$ a mass density). Moreover, $\mathcal{E}$ is an extensive, global quantity, and even if $\rho \mathcal{E}$ was an energy density, would it have a physical meaning ? And if so, what meaning ? – Georg Sievelson Feb 12 '13 at 1:15
As to the etymology, as the enstrophy is the integral of the vorticity squared, I expected that it was related to rotation, so it does not help me that muche, but it's nice to know. – Georg Sievelson Feb 12 '13 at 1:19
@GeorgSievelson I am sorry I misinterpreted your question. Looking at the integral in your question, it is not clear you are talking about a D-dimensional space. I also thought that you were looking for an interpretation of the word, as you said "It has a name, it must mean something". I have edited my answer, if you wish to read it. – JKL Feb 12 '13 at 9:12

The Wikipedia article you have sited states a simple but clear interpretation of this quantity and some basic use:

"The enstrophy can be interpreted as another type of potential density (ie. see probability density); or, more concretely, the quantity directly related to the kinetic energy in the flow model that corresponds to dissipation effects in the fluid. It is particularly useful in the study of turbulent flows, and is often identified in the study of thrusters as well as the field of flame theory."

This seems to answer your basic queries. However, I understand from the previous answer that you want a bit more information on this quantity and its use. Well...

I don't believe I can define enstrophy any more clearly that you have yourself (and Wikipedia does) already, however, let’s try, I can then give you some idea of its use. Local anisotropic behaviour of enstrophy can be treated as the self-organization of the regions of high vorticity in coherent vortex structures – most notably vortex filaments/tubes. This turbulent behaviour is ubiquitous and still not very well understood. In particular, local alignment or anti-alignment of the vorticity direction, i.e. local coherence, is prominently featured in turbulent flows.

It is thought, but still an area of active research, that evolution of voticity in turbulent flows leads/is governed by 3D enstrophy cascades on 'large-scales' of the turbulence. I am no expert in this area but have read a few papers on this when writing a 2D code for turbulence modelling.

I hope this helps.

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