Modelling and calculation for shear-driven rotating turbulence, with multiscale and directional approach

Abstract : Stability and turbulence in rotating shear flows is essential in many contexts ranging from engineering—as in e.g. turbomachinery or hydraulic energy production—to geophysics and astrophysics. Apart from inhomogeneous effects which we discard in the present study, these flows are complex because they involve an anisotropic dynamics which is difficult to represent at the level of one-point statistics. In this context, the properties of these flows, such as scale-by-scale anisotropy or turbulent cascade can be studied via two-point statistical models of Homogeneous Anisotropic Turbulence (HAT), in which the distorting mean flow is represented by uniform mean velocity and density gradients, and by body forces as the Coriolis one. The context of HAT can be relevant for flows in a plane channel with spanwise rotation, or for a Taylor-Couette flow. We propose a new model for predicting the dynamics of homogeneous sheared rotating turbulence. The model separates linear distortion effects from nonlinear turbulent dynamics, so that each contribution can be treated with an adapted model. Our model deals with equations governing the spectral tensor of two-point second-order velocity correlations, and is developed for arbitrary mean velocity gradients with or with- out system rotation. The direct linear effect of mean gradients is exact in our model, whereas nonlinear effects come from two-point third-order correlations which are closed by an anisotropic EDQNM model. In the closure, the anisotropy is restricted to an expansion in terms of low-degree angular harmonics (Mons et al., 2016). The present model has been validated in the linear regime, by comparison to the accurate solution of viscous Rapid Distortion Theory (vRDT), in several cases, stabilizing, destabilizing or neutral. In contrast with pseudo-spectral DNS adapted to shear flow by Rogallo (1981) in en- gineering and by Lesur & Longaretti (2005) in astrophysics, the advection operator is not solved by following characteristic lines in spectral or physical space, but by an original high- order finite-difference scheme for calculating derivatives ∂ i with respect to the wave vector k. One thus avoids mesh deformation and remeshing, thus one can easily extract angular ii harmonics at any time since physical or spectral space are not distorted. With this new approach, we are able to improve the prediction of the previous model by Mons et al. (2016), in which the linear resolution is questioned at large time, especially in the case without rotation. The proposed new model is versatile since it is implemented for several cases of mean velocity gradients consistent with the homogeneity approximation. Validations have been done for several cases of plane deformations. In the case of sheared turbulence, whose modelling resists most one-point approaches and even the two-point model by Mons, we propose an adaptation of our two-point model in a new hybrid model, in which return-to- isotropy is explicitly introduced in the guise of Weinstock (2013)’s model. Predictions of the new hybrid model are extremely good.
Keywords : Shear flows
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Ying Zhu. Modelling and calculation for shear-driven rotating turbulence, with multiscale and directional approach. Other. Université de Lyon, 2019. English. ⟨NNT : 2019LYSEC002⟩. ⟨tel-02161933⟩

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