Effects of Large Scales Motion in Unstable Stratified Shear Flows

Abstract

Homogeneous turbulence under unstable uniform stratification (N ² < 0) and vertical shear $(S = \hbox{d}\overline{U}_{1}/\hbox{d}x_3)$ is investigated by using the linear theory (or the so-called rapid distortion theory, RDT) for an initial isotropic turbulence over a range −∞ ≤ R _i =N ²/S ² ≤ 0. The initial potential energy is zero and P _r =1 (i.e. the molecular Prandtl number).

One-dimensional (streamwise) k ₁−spectra, especially Θ₃₃(k ₁) (i.e., that of the vertical kinetic energy, $\overline{v_{3}v_{3}}/2)$ are investigated. In agreement with previous experiments, it is found that the unstable stratification affects the turbulence quantities at all scales. A significant increase of the vertical kinetic energy is observed at low wavenumbers k ₁ (i.e. large scales) due to an increase of the stratification $\left(\sqrt{-N^{2}}\right)$. The effect of the shear (S) is appreciable only at high wavenumbers k ₁ (i.e. small scales).

Based on the importance of the spectral components with k ₁ = 0, the asymptotic forms of Θ _ij (k ₁ = 0) or equivalently the so-called “two-dimensional” energy components (2DEC) are analyzed in detail. The asymptotic form for the ratio of 2DEC is compared to the long-time limit of the ratio of real energies. In the unstable stratified shearless case (S=0,N ² ≠ 0) the variances and the covariances of the velocity and the density are derived analytically in terms of the Weber functions, while when S ≠ 0 and N ² ≠ 0 they are obtained numerically (−100 ≤ R _i <0 and $t^{+} = \sqrt{S^{2}-N^{2}} t = 100)$. The results are discussed in connection to previous experimental results in unstable stratified open channel flows cooled from above by Komori et al. Phy Fluids 25, 1539–1546 (1982).

It is shown that the Richardson number dependence of the long-time limit of the ratios of real energies is well described by this “simple” model (i.e. the dependence of the long-time limit of 2DEC on R _i ). For example, the ratio of the potential energy to the kinetic energy (q ²/2), approaches −R _i /(1−R _i ), the ratio of turbulent energy production by buoyancy forces to production by shearing forces (i.e. the flux Richardson number, R _if ), approaches R _i . Also, the Richardson number dependence of the principal angle (β) of the Reynolds stress tensor and the angle (β_ρ) of the scalar flux vector is fairly predicted by this model $\beta ,\beta_{\rho} \rightarrow 1/2\tan^{-1}\left[2\sqrt{-R_{i}}/(1+R_i)\right]$.

On the other hand, it is found that the above ratios are insensitive to viscosity, while the ratios ɛ /q ² and $\varepsilon_{\rho}\left/(2PE)\right.$, depend on the viscosity and they evolve asymptotically like t ⁻¹. The turbulent Froude number, F _rt =(L _Oz /L _E )^2/3, where L _Oz and L _E are the Ozmidov length scale and the Ellison length scale, respectively, evolves asymptotically like t ^−1/3.