Abstract
This chapter is devoted to the pure shear case. Main results of Reynolds Stress Models are given, and the specificity of Rapid Distortion Theory is investigated for transient growth, in terms of poloidal/ toroidal modal decomposition. The scale-by-scale return to isotropy is addressed, with the support of the simplified model in terms of spherically averaged descriptors. New DNS results are exploited to show a quasi-balance between spectral transfer and dissipation transfer, for largest scales, at Reynolds numbers much smaller than those in decaying isotropic turbulence. Finally, a discussion of regeneration cycles and self-sustaining process is given.
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Notes
- 1.
In isotropic turbulence, all these quantities reduce to a single one, say \(L_f\), with \(L^{(n)}_{nn} = L_f\) (any n, no summation on it), \(L^{(n)}_{ii} = L_f/2\) if \(n\ne i\) (no summation on i) and \(L^{(n)}_{ij}=0\) if \(i \ne j\). Accordingly, departure from this simple relationship reflects anisotropic structure too.
- 2.
It is recalled that in this case one has \(\det \mathsf{F} =1\).
- 3.
Hamiltonian formalism is also very important in the “Russian” school of wave turbulence theory.
- 4.
Confusing notations in the original paper have been corrected here, as far as possible; the EDQNM-like integral uses a polar-spherical system of coordinates for \({\varvec{p}}\) with polar axis \(\varvec{k}\). Formulation in terms of the bipolar system of coordinates (as in Chap. 4) is easily recovered from
$$\iiint (...) d^3 {\varvec{p}}= 2 \pi \int ^{\infty }_0 p^2 dp \int ^{\pi }_0 (...) sin \theta _p d \theta _p = 2 \pi \iint _{\Delta _k} \frac{pq}{k} ( ...) dp dq.$$ - 5.
- 6.
Such a simplification can be obtained in several ways. First, assuming that velocity fluctuations are nearly Gaussian, third-order moments are identically null. Second, looking at the structure of this term, it is seen that the difference between the two third-moment terms vanishes in isotropic turbulence. Then, assuming that we are dealing with weak departure from isotropy in the inertial range, this term can be assumed to be small in front of the other ones.
- 7.
It is recalled here that an external length scale, other than those related to the fluctuating flow statistics, is missing in strictly homogeneous turbulence.
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Sagaut, P., Cambon, C. (2018). Incompressible Homogeneous Anisotropic Turbulence: Pure Shear. In: Homogeneous Turbulence Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-73162-9_9
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