Abstract
This chapter aims to convey the underlying theoretical concepts and the current state of research in the field of turbulent multi-particle dispersion. Some of these concepts are well known in the turbulence community and are described in detail in popular textbooks. Wherever possible, however, I will rely on the excellent and historically important texts by Richardson, Kolmogorov and Batchelor in order to present these ideas. Section 2.1 describes the governing equations of a turbulent flow and explains the necessity of a statistical description. Section 2.2 focuses on the famous theory of Kolmogorov and the underlying picture of the turbulence energy cascade. In Sect. 2.3, different aspects of turbulent dispersion for two or more fluid particles are described and current theoretical and experimental findings are presented.
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- 1.
The smallest turbulent scale is estimated from measurements in atmospheric clouds at an altitude of approx. 2000 m (Siebert et al. 2006). The atmospheric pressure at this height is about 65 kPa for a sea-surface temperature of 288 K and an average molecular mass of the atmosphere of 0.02897 kg/mole, leading to a mean free path of \(\lambda \approx {10^{-7}}\) m.
- 2.
An eddy is to be understood as “a component of motion with a certain length scale, i.e. an arbitrary flow pattern characterized by size alone” (Batchelor 1950). It is not to be confused with a whirl as seen near a drain.
- 3.
- 4.
- 5.
One can also think of \(t_0\) as the eddy-turn-over time of an eddy of size \(R_0\). After one turn-over, the eddy looses its coherence and the velocities of two fluid particles belonging to the eddy become uncorrelated.
- 6.
For an incompressible flow, it can be shown that the displacement probability of two particles obeys \(P(\mathbf {x}_1,\mathbf {x}_2, t| \mathbf {y}_1,\mathbf {y}_2,0) = P(\mathbf {y}_1,\mathbf {y}_2,0|\mathbf {x}_1,\mathbf {x}_2, t)\) (Lundgren 1981), which directly leads to \(P_r(\mathbf {R}, t|\mathbf {R}_0,0)=P_r(\mathbf {R}_0,0|\mathbf {R}, t)\).
- 7.
Instead of the shape tensor, one can also define the dispersion tensor \(\mathbf {C}(t)=\mathrm {P}^T(t) \mathrm {P}(t) = \mathbf {W}(t)\, \text {diag}(\sigma _1^2, \sigma _2^2, \ldots , \sigma _{\text {min}(d,n-1)}^2) \mathbf {W}^T(t) \). It has the same eigenvalues as \(\mathbf {G}(t)\) and serves the same purpose (Hackl et al. 2011).
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Jucha, J. (2015). Introduction and Theory. In: Time-Symmetry Breaking in Turbulent Multi-Particle Dispersion. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-19192-8_2
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