Abstract
This chapter presents the specific features of the description of compressible turbulent flows: governing equations, Favre-averaging and Kovasznay decomposition of compressible disturbances as the sum of vortical, acoustic and entropy modes, along with Rankine–Hugoniot jump relations. Linearized jump relations are also discussed.
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Notes
- 1.
- 2.
The discussion in the present chapter is restricted to the case of single-phase, non-reactive, single-species flows of divariant fluids. The case of binary mixtures of perfect gas is addressed in Sect. 16.9.2.
- 3.
It is recalled that the first non-dimensional parameter \(\epsilon \) is related to the amplitude of the perturbations.
- 4.
Considering \(\nu _0 = 0.15\cdot 10^{-4}\,\mathrm{m^2 s}^{-1}\) and \(a_0 = 300\,\mathrm{m s}^{-1}\), one obtains \(\nu _0/a_0 = 5 \cdot 10^{-8}\,\mathrm{m}\) which is of the order of the mean-free path of the molecules in the gas. For a frequency equal to 1 Hz, one has \(1/k = 300\,\mathrm{m}\) and \(\epsilon ' =1.66 \cdot 10^{-10}\). For 1 kHz one has \(\epsilon ' =1.66 \cdot 10^{-7}\) and \(\epsilon ' =1.66 \cdot 10^{-4}\) at 1 MHz. Even at 1 GHz, one obtains \(\epsilon '=1.66 \cdot 10^{-1} < 1\) !
- 5.
But let us notice that the density-weighted average was introduced by Osborne Reynolds in is seminal paper in 1884!
- 6.
This term was coined by Chassaing and coworkers (see Chassaing et al. 2002), who developed the alternative ternary regrouping approach.
- 7.
This assumption is relevant for most high-speed non-reactive flows.
- 8.
In non-homogeneous flows a third contribution must be taken into account, which is defined as
$$ {\bar{\varepsilon }}_n = 2 \frac{{\bar{\mu }}}{{\bar{\rho }}} \left( \frac{\partial ^2 }{\partial x_i \partial x_j} \overline{u'_iu'_j} - 2 \frac{\partial }{\partial x_j} \left( \overline{ {u}^{\prime }_j \frac{\partial {u}^{\prime }_i}{\partial x_i} } \right) \right) . $$
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Sagaut, P., Cambon, C. (2018). Additional Reminders: Compressible Turbulence Description. In: Homogeneous Turbulence Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-73162-9_3
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DOI: https://doi.org/10.1007/978-3-319-73162-9_3
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