From Vorticity to Velocity

As we have already seen earlier on, to obtain the velocity field from the vorticity one we have to solve the following equation:

$ \nabla^2 \mathbf{u} = - \nabla \times \boldsymbol{\omega} $

Since the Green function for the Poisson equation is: $ G(\mathbf{x})=(4 \pi |\mathbf{x}|)^{-1} $ we obtain:

$ \mathbf{u}(\mathbf{x})= - \int{ \frac{1}{4 \pi (\mathbf{x} - \mathbf{y}) } \nabla \times \boldsymbol{\omega}(\mathbf{y})} $

which becomes after few steps:

$ \mathbf{u}(\mathbf{x})= - \frac{1}{4 \pi} \int{ \frac{\mathbf{x} - \mathbf{y}}{ (\mathbf{x}-\mathbf{y})^3 } \times \boldsymbol{\omega}(\mathbf{y})} $.

So far we have applied the SEM to the vorticity and then calculated the velocity field. In particular, the structure created were isotropic:

$ \boldsymbol{\omega}'(\mathbf{x})=\sqrt{\frac{1}{N}}\sum_{k=1}^N\boldsymbol{\alpha}^k g_\sigma(\frac{ | \mathbf{x} - \mathbf{x}^k |}{\sigma}) $

What if we applied a modified SEM:

$ \boldsymbol{\omega}'(\mathbf{x})=\sqrt{\frac{1}{N}}\sum_{k=1}^N\boldsymbol{\alpha}^k g_x(\frac{ | x - x^k | }{\sigma_x}) g_y(\frac{ | y - y^k | }{\sigma_y}) g_z(\frac{ | z - z^k | }{\sigma_z})$

We get then:

$ \mathbf{u}(\mathbf{x})= - \frac{1}{4 \pi} \int{ \frac{\mathbf{x} - \mathbf{y}}{ (\mathbf{x}-\mathbf{y})^3 } \times \sqrt{\frac{1}{N}}\sum_{k=1}^N\boldsymbol{\alpha}^k g_x(\frac{ | x - x^k | }{\sigma_x}) g_y(\frac{ | y - y^k | }{\sigma_y}) g_z(\frac{ | z - z^k | }{\sigma_z})} $.

$ \mathbf{u}(\mathbf{x})= - \frac{1}{4 \pi} \sqrt{\frac{1}{N}}\sum_{k=1}^N \int{ \frac{\mathbf{x} - \mathbf{y}}{ (\mathbf{x}-\mathbf{y})^3 } \times \boldsymbol{\alpha}^k g_x(\frac{ | x - x^k | }{\sigma_x}) g_y(\frac{ | y - y^k | }{\sigma_y}) g_z(\frac{ | z - z^k | }{\sigma_z})} dx dy dz $

Integration of the formula stated

I am defining a matlab script which should integrate the u equation!!


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Topic revision: r4 - 2010-11-15 - 11:32:33 - RuggeroPoletto
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