# Definitions for enstrophyen·stro·phy

### WiktionaryRate this definition:1.0 / 1 vote

1. enstrophynoun

A measure of the kinetic energy of a fluid as a result of turbulence.

### WikipediaRate this definition:0.0 / 0 votes

1. Enstrophy

In fluid dynamics, the enstrophy E can be interpreted as another type of potential 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 combustion theory. Given a domain Ω ⊆ R n {\displaystyle \Omega \subseteq \mathbb {R} ^{n}} and a once-weakly differentiable vector field u ∈ H 1 ( R n ) n {\displaystyle u\in H^{1}(\mathbb {R} ^{n})^{n}} which represents a fluid flow, such as a solution to the Navier-Stokes equations, its enstrophy is given by: E ( u ) := ∫ Ω | ∇ u | 2 d x , {\displaystyle {\mathcal {E}}(u):=\int _{\Omega }|\nabla \mathbf {u} |^{2}\,dx,} where | ∇ u | 2 = ∑ i , j = 1 n | ∂ i u j | 2 {\displaystyle |\nabla \mathbf {u} |^{2}=\sum _{i,j=1}^{n}\left|\partial _{i}u^{j}\right|^{2}} . This quantity is the same as the squared seminorm | u | H 1 ( Ω ) n 2 {\displaystyle |\mathbf {u} |_{H^{1}(\Omega )^{n}}^{2}} of the solution in the Sobolev space H 1 ( Ω ) n {\displaystyle H^{1}(\Omega )^{n}} . In the case that the flow is incompressible, or equivalently that ∇ ⋅ u = 0 {\displaystyle \nabla \cdot \mathbf {u} =0} , the enstrophy can be described as the integral of the square of the vorticity ω {\displaystyle \mathbf {\omega } } , E ( ω ) ≡ ∫ Ω | ω | 2 d x {\displaystyle {\mathcal {E}}({\boldsymbol {\omega }})\equiv \int _{\Omega }|{\boldsymbol {\omega }}|^{2}\,dx} or, in terms of the flow velocity, E ( u ) ≡ ∫ S | ∇ × u | 2 d S . {\displaystyle {\mathcal {E}}(\mathbf {u} )\equiv \int _{S}|\nabla \times \mathbf {u} |^{2}\,dS\,.} In the context of the incompressible Navier-Stokes equations, enstrophy appears in the following useful result d d t ( 1 2 ∫ Ω | u | 2 ) = − ν E ( u ) . {\displaystyle {\frac {d}{dt}}\left({\frac {1}{2}}\int _{\Omega }|\mathbf {u} |^{2}\right)=-\nu {\mathcal {E}}(\mathbf {u} ).} The quantity in parentheses on the left is the energy in the flow, so the result says that energy declines proportional to the kinematic viscosity ν {\displaystyle \nu } times the enstrophy.

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1. Enstrophy

In fluid dynamics, the enstrophy can be described as the integral of the square of the vorticity given a velocity field as, Here, since the curl gives a scalar field in 2-dimensions corresponding to the vector-valued velocity solving in the incompressible Navier–Stokes equations, we can integrate its square over a surface S to retrieve a continuous linear operator on the space of possible velocity fields, known as a current. This equation is however somewhat misleading. Here we have chosen a simplified version of the enstrophy derived from the incompressibility condition, which is equivalent to vanishing divergence of the velocity field, More generally, when not restricted to the incompressible condition, or to two spatial dimensions, the enstrophy may be computed by: where is the Frobenius norm of the gradient of the velocity field . The enstrophy can be interpreted as another type of potential 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.

### Numerology

1. Chaldean Numerology

The numerical value of enstrophy in Chaldean Numerology is: 4

2. Pythagorean Numerology

The numerical value of enstrophy in Pythagorean Numerology is: 5

### Popularity rank by frequency of use

enstrophy#100000#307348#333333

## Translations for enstrophy

• enstrophy
• Enstrophie
• énstrophie
• एंस्ट्रोफी
• enstrophy
• энстрофия

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