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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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.
References
Arneodo, A., Manneville, S. & Muzy, J. F. (1998). Towards log-normal statistics in high Reynolds number turbulence. European Physical Journal B, 1, 129–140.
Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F. & Succi, S. (1993). Extended self-similarity in turbulent flows. Physical Review E, 48, R29.
Castaing, B. (1989). Conséquences d’un principe d’extremum en turbulence. Journal of Physique, 50, 147.
Castaing, B. (1996). The temperature of turbulent flows. Journal of Physique 2, 6, 105–114.
Chapman, C. J. & Proctor, M. R. E. (1980). Nonlinear Rayleigh–Benard convection between poorly conducting boundaries. Journal of Fluid Mechanics, 101, 759–782.
Cheng, X.-L., Wang, B.-L. & Zhu, R. (2010). Kolmogorov Constants of Atmospheric Turbulence over a Homogeneous Surface. Atmospheric and Oceanic Science Letters, 3, 195–200.
Chorin, A. (1991). Vorticity and turbulence. New York: Springer.
Davidson, P. (2004). Turbulence: An introduction for scientists and engineers. Oxford: Oxford University Press.
Dubrulle, B. (1994). Intermittency in fully developed turbulence: Log-Poisson statistics and generalized scale covariance. Physical Review Letters, 73, 959.
Frisch, U. (1991). From global scaling, à la Kolmogorov, to local multifractal scaling in fully developed turbulence. Proceedings of the Royal Society of London, 434, 89.
Frisch, U. 1995 Turbulence: The legacy of A. N. Kolmogorov. Cambridge: Cambridge University Press.
Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A. (2003). Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Physics of Fluids, 15, L21–L24.
von Kármán, T. & Howarth, L. (1938). On the statistical theory of isotropic turbulence. Proceedings of the Royal Society of London, 164A, 192.
Launder, B. E. & Spalding, D. B. (1972). Lectures in mathematical models of turbulence. London: Academic Press.
Lesieur, M. (1990). Turbulence in fluids. Dordrecht: Kluwer.
Leslie, D. (1973). Developments in the theory of turbulence. Oxford: Oxford Science Publishier.
McComb, W. D. (1990). The physics of fluid turbulence. Oxford: Oxford Science Publishier.
Monin, A. & Yaglom, A. (1975). Statistical fluid mechanics (Vol. 1). Cambridge: MIT Press.
Orszag, S. (1973). Fluids Dynamics. In Balian, R., & Peube, J.L. (Eds), Lectures on the statistical theory of turbulence. (pp. 237–374), Gordon and Breach, New York, 1977.
Piquet, J. (2001). Turbulent flows: Models and physics. New York: Springer.
Prandtl, L. (1925). Bericht über Untersuchungen zur ausgebildeten Turbulenz. Zeit. Ang. Math. Mech., 5, 136–139.
She, Z.-S. & Lévêque, E. (1994). Universal scaling laws in fully developed turbulence. Physical Review Letters, 72, 336.
She, Z.-S. & Waymire, E. (1995). Quantized energy cascade and Log-Poisson statistics in fully developed turbulence. Physical Review Letters, 74, 262.
Smagorinsky, J. (1963). General circulation experiments with the primitive equations. I. The basic experiment. Monthly Weather Review, 91, 99.
Tennekes, H. & Lumley, J. (1972). A first course in turbulence. Cambridge: MIT Press.
Vincent, A. & Meneguzzi, M. (1991). The spatial structure and statistical properties of homogeneous turbulence. Journal of Fluid Mechanics, 225, 1–20.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Rieutord, M. (2015). Turbulence. In: Fluid Dynamics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-09351-2_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-09351-2_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09350-5
Online ISBN: 978-3-319-09351-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)