Divergence Helmholtz

The initial conditions for the velocity components are set up so that there is a tubular shear layer aligned along the 2 -direction at time t = 0. The tv-velocity has a top-hat profile with a tan-hyperbolic shear layer. Stream wise and azimuthal perturbations are introduced to expedite roll-up and the development of the Widnall instability. The details can be found in [7]. The initial velocity field is made divergence-free using the Helmholtz decomposition. The size of the computational domain (one periodic cubical box) is 4do on each side. 

Taking the divergence of Equation (35) and substituting it into Equation (37), a second-order partial differential equation of the Helmholtz type is finally obtained for the PI approximation... 

The forth equation in (8.62) can be obtained by applying the divergence operator V to both sides of the first equation and allowing for the continuity equation for the extraneous currents and charges. In particularly, the Helmholtz equations (8.31) can be written as follows... 

Helmholtz analysis. So, a vector field with the desired curl is evaluated and projected in the space of divergence-free vectors. This concept leads to the next assertion ... 

Helmholtz s theorem states that A general vector field, that vanishes at infinity, can be completely represented as the sum of two independent vector fields, one that is irrotational (zero curl) and another that is solenoidal (zero divergence) [Morse and Feshbach, 1953 Plonsey and CoUin, 1961]. The impressed current density P is a vector field that vanishes at infinity and, according to the theorem, maybe expressed as the sum of two components ... 

In the mid-nineteenth century, several fields of science saw explosive activity. Men like Orstead, Gauss, Weber, Laplace, Lagrange, Camot, Faraday, Maxwell, and Helmholtz were contemporaries in physics and chemistry. Divergence and segregation of the fields gradually occurred at this time, but some workers, such as Helmholtz, crossed these boundaries. 

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