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I Spectral representation of an incompressible flow

  1. Nov 27, 2016 #1
    Hi PH.

    Let ##u_i(\mathbf{x},t)## be the velocity field in a periodic box of linear size ##2\pi##. The spectral representation of ##u_i(\mathbf{x},t)## is then
    $$u_i(\mathbf{x},t) = \sum_{\mathbf{k}\in\mathbb{Z}^3}\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j}$$ where ι denotes the usual imaginary unit.

    A flow is incompressible iff ##u_i(\mathbf{x})## is a solenodial verctor field, that is
    $$\nabla_iu_i(\mathbf{x},t) = 0$$
    Combining the above, I get
    $$\begin{aligned}0 &= \nabla_iu_i(\mathbf{x},t) \\
    &= \sum_{\mathbf{k}\in\mathbb{Z}^3}\hat{u}_i(\mathbf{k},t)\nabla_ie^{\iota k_jx_j} \\
    &= \sum_{\mathbf{k}\in\mathbb{Z}^3}\iota k_j\delta_{ij}\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j} \\
    &= \iota\sum_{\mathbf{k}\in\mathbb{Z}^3}k_j\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j}\end{aligned}$$

    If ##u_i(\mathbf{x},t)## is to represent a physical flow velocity field, then ##u_i(\mathbf{x},t)## must be real function. That is ##\mathbf{u}(\mathbf{x}):[0,2\pi]^3\times\mathbb{R}\rightarrow\mathbb{R}^3## and thus ##\hat{u}_i(\mathbf{k}) = \hat{u}^\ast_i(-\mathbf{k})##. Therefore
    $$\begin{aligned} \iota\sum_{\mathbf{k}\in\mathbb{Z}^3}k_j\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j} &= -\iota\sum_{\mathbf{k}\in\mathbb{N}^3}k_j\hat{u}_i(-\mathbf{k},t)e^{-\iota k_jx_j} + 0 + \iota\sum_{\mathbf{k}\in\mathbb{N}^3}k_j\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j} \\
    &= \iota\sum_{\mathbf{k}\in\mathbb{N}^3}k_j\big[\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j} - \hat{u}_i(-\mathbf{k},t)e^{-\iota k_jx_j}\big] \\
    &= \iota\sum_{\mathbf{k}\in\mathbb{N}^3}k_j\big[\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j} - \hat{u}^\ast_i(\mathbf{k},t)\big(e^{\iota k_jx_j}\big)^\ast\big] \\
    &= 2\iota\sum_{\mathbf{k}\in\mathbb{N}^3}\Im\big[k_j\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j}\big] \end{aligned}$$

    From the harmonic analysis I would expect the incompressibility condition to be $$k_i\hat{u}_i(\mathbf{k},t) = 0$$
    Clearly, if this is true the incompressibility condition would be true. But I can't see how (unless I assume a sort of detailed balance in the above sum) the incompressibility condition imply the orthogonal relation ##k_i\hat{u}_i(\mathbf{k},t) = 0##.

    Any help is appreciated :-)
     
    Last edited: Nov 27, 2016
  2. jcsd
  3. Dec 2, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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