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Eulerian transverse velocity increment

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Old   May 6, 2020, 23:36
Default Eulerian transverse velocity increment
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I am reading paper [1], and I run into this formula for the Eulerian transverse velocity increment

\delta u_\perp = \|\mathcal{P}\cdot\delta u\|\cos\theta

where \mathcal{P} = \delta_{ij} - \hat{r}_i\hat{r}_j is the projection tensor projecting the vector onto the plane perpendicular to the vector r, and \forall\theta\in[0\,,2\pi], and \hat{r} simply being the normalized vector of r.

I understand the projection, but I don't understand why multiplied by a \cos\theta? Can anyone shed some light?

Appreciate it.

[1]. E. Leveque, A. Naso. Introduction of longitudinal and transverse Lagrangian velocity increments in homogeneous and isotropic turbulence. EPL, 108, 54004, 2014.
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Old   May 7, 2020, 09:45
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The projection involves the dot product. The cos term results from that.
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turbulence analysis, velocity increments

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