Abstract
As we pointed it out in the first pages of this book, the understanding of turbulence remains one of the challenges of nowadays physics. The goal of this chapter is to introduce the reader to the main approaches that are used to deal with this difficult problem.
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Notes
- 1.
Let us note here that the true stress induced by the correlation \(\left \langle \mathbf{v}' \otimes \mathbf{ v}'\right \rangle\) is rather − R ij since the momentum equation (9.7) may also be written
$$\displaystyle{\rho \frac{D\langle v_{i}\rangle } {\mathit{Dt}} = \partial _{j}\sigma _{\mathit{ij}}}$$with \(\sigma _{\mathit{ij}} = -\left \langle P\right \rangle +\mu (\partial _{i}\langle v_{j}\rangle + \partial _{j}\langle v_{i}\rangle ) - R_{\mathit{ij}}\). Note also that the Reynolds tensor is often defined as \(\left \langle \mathbf{v}' \otimes \mathbf{ v}'\right \rangle\).
- 2.
A pseudo-scalar is a scalar quantity the sign of which depends on the orientation of the vector basis. For instance, the determinant of three vectors (in three dimensions) is a pseudo-scalar. In our case, if \(\mathbf{X}\) et \(\mathbf{Y }\) are two vectors, from the definition of [Q], X i Y j Q ij is a true scalar. Thus
$$\displaystyle{A(\mathbf{X} \cdot \mathbf{ Y })^{2} + (\mathbf{r} \cdot \mathbf{ X})(\mathbf{r} \cdot \mathbf{ Y })B/r^{2} + H\epsilon _{\mathit{ ijk}}X_{i}Y _{j}r_{k}/r}$$is a true scalar. In this expression we see that the last term is the determinant of three vectors times H. Thus H is a pseudo-scalar.
- 3.
First experimental values as those given by Monin and Yaglom (1975) are around 1.5. Recent measurements in the atmospheric boundary layer by Cheng et al. (2010) give 1.56. Numerical experiments have long given values around 2 (e.g. Vincent and Meneguzzi 1991), but recently it has been understood that the numerical resolution was an important issue. The latest results obtained with the very high resolution numerical simulations are getting closer to experimental values (Kaneda et al. 2003).
- 4.
However, this assumption is still approximate because there is no good reason that fluctuations towards small values are as probable as those towards high values.
- 5.
We should keep in mind that in 1962, the Kolmogorov spectrum had already been observed experimentally, and thus any new theory should reproduce this result.
- 6.
This is the turbulence which appears in the wake of a grid. It is homogeneous in the directions parallel to the grid
- 7.
We saw that the exponent ζ 2 was related to exponent of the energy density spectrum. The change implied by this new theory compared to the Kolmogorov one is very small: this exponent is now: \(-\frac{5} {3} - 0.03\).
- 8.
A function f is concave , if the following inequality f[(x + y)∕2] ≤ (f(x) + f(y))∕2 is verified. For a continuous and derivable function, this inequality is equivalent to f″(x) ≥ 0.
- 9.
Markovian processes are such that the probability of an event does not depend on the history of the process.
- 10.
Let us mention that usually subgrid scale models are not categorized in models of turbulence since they give a local prescription that can be used only in numerical simulation. However, their similarity with the mean-field approach is strong enough that we discuss them here.
- 11.
We noted that \(\sigma '_{\mathit{ij}} =\nu (\partial _{j}v'_{i} + \partial _{i}v'_{j})\).
- 12.
See appendix for the demonstration.
- 13.
Indeed, the local properties of turbulence can only be, with this model, characterized by the two scalars \(K\) and \(\varepsilon\). In the present case K is the only one dimensionly correct.
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Rieutord, M. (2015). Turbulence. In: Fluid Dynamics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-09351-2_9
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