Abstract
Flows at low Reynolds number are generally regular, which are known as laminar flows. As the Reynolds number increases, the flow becomes more spatially and temporally irregular. Such flow is called a turbulent flow. Turbulent flows are characterized by unsteady three-dimensional motion, and large energy dissipation ability. A precise description of turbulent flows is impossible, but the characteristics of irregular phenomena exhibit some degree of statistical regularity because of their randomness. In this chapter, we discuss turbulence in terms of its statistical characteristics such as average velocity, dispersion, and correlation. This chapter consists of seven topics as follows: (1) Transition from laminar flow to turbulent flow, (2) Stability theory of flow, (3) Basic equation of turbulent flow, (4) Closure, (5) Shear-flow Turbulence near a wall (ground), (6) Law of similarity in homogeneous isotropic turbulence, and (7) Turbulent flow analysis.
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Notes
- 1.
Strictly speaking, the flow can become unstable when \(\mathrm {Im}(\omega ) = 0\). If \(\mathrm {Im}(\omega ) = 0\) and multiple solutions exist, the disturbance develops in a linear fashion.
- 2.
Most textbooks provide more systematic arguments using velocity correlation tensors. Here we limit our argument to the minimum requirement for the energy Eq. (5.84) required by the law of similarity. Readers who are unfamiliar with this topic may nonetheless find the calculations overly complicated. Such readers are recommended to skip Sects. 5.6.2 and 5.6.3 (but should read the conclusions to this section (5.84) and the sections beyond Sect. 5.6.4.
- 3.
Isotropic turbulence is homogeneous by definition, and is therefore sometimes simply called isotropic turbulence.
- 4.
As we here consider an infinite region, V and \(\int u^2 dr\) are also infinite. Strictly speaking, this situation should be treated by limiting the range of \((1/2V) \int u^2 dr\) and taking the limit as \(V \rightarrow \infty \).
- 5.
Note that here we divide by S, not V, because we are dealing with two-dimensional volumes. Of course, the integration over \(d\varvec{r}\) is similarly performed in two dimensions.
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Department of Earth System Science and Technology, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University. (2017). Turbulent Flow. In: Fluid Dynamics for Global Environmental Studies. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56499-7_5
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DOI: https://doi.org/10.1007/978-4-431-56499-7_5
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