The development of turbulence is investigated in the presence of a mean plane shear flow (rate S) rotating with angular velocity vector (rate Ω) perpendicular to its plane. An important motivation was generalizing the work by Lee, Kim & Moin (1990) to rotating shear flow, in particular detailed comparisons of homogeneous rapid distortion theory (RDT) results and the databases of homogeneous and channel flow direct numerical simulations (DNS). Linear analysis and related RDT are used starting from the linearized equations governing the fluctuating velocity field. The parameterization based on the value of the Bradshaw–Richardson number B=R(1+R) (with R=−2Ω/S) is checked against complete linear solutions. Owing to the pressure fluctuation, the dynamics is not governed entirely by the parameter B, and the subsequent breaking of symmetry (between the R and −1 −R cases) is investigated. New analytical solutions for the ‘two-dimensional energy components’ [Escr ](l)ij =Eij(kl=0, t) (i.e. the limits at kl=0 of the one-dimensional energy spectra) are calculated by inviscid and viscous RDT, for various ratios Ω/S and both streamwise l=1 and spanwise l=3 directions. Structure effects (streak-like tendencies, dimensionality) in rotating shear flow are discussed through these quantities and more conventional second-order statistics. In order to compare in a quantitative way RDT solutions for single-point statistics with available large-eddy simulation (LES) data (Bardina, Ferziger & Reynolds 1983), an ‘effective viscosity’ model (following Townsend) is used, yielding an impressive agreement.
The essentials of rapid distortion theory (RDT) are briefly recalled for homogeneous turbulence subjected to rotational mean flows, including its linkage to stability analysis. The latter “linkage” is of particular importance from our viewpoint, since it also attracted the attention of Charles Speziale, resulting in at least two papers [Speziale, C. G., Abid, R., and Blaisdell, G. A., 1996, “On the Consistency of Reynolds Stress Turbulence Closures With Hydrodynamic Stability Theory,” Phys. Fluids, 8, pp. 781–788 and Salhi, A., Cambon, C., and Speziale, C. G., 1997, “Linear Stability Analysis of Plane Quadratic Flows in a Rotating Frame,” Phys. Fluids, 9(8), pp. 2300–2309] with particular emphasis on rotating flows. New analytical solutions and related RDT results are presented for shear flows including buoyancy forces, with system rotation or mean density stratification. Finally, combining shear, rotation and stratification, RDT is shown to be pertinent to revisiting the baroclinic instability. This instability results from the tilting of mean isopycnal surfaces under combined effects of vertical shear and system rotation, in a vertically (stably) stratified medium rotating around the vertical direction. In addition, the challenge of reproducing RDT dynamics in single-point closure models is briefly discussed, from the viewpoint of structure-based modeling [Cambon C., Jacquin, L., and Lubrano, J.-L., 1992, “Towards a New Reynolds Stress Model for Rotating Turbulent Flows,” Phys. Fluids A, 4, pp. 812–824 and Kassinos, S. C., Reynolds, W. C., and Rogers, M. M., 2000, “One-Point Turbulence Structure Tensors,” J. Fluid Mech., 428, pp. 213–248.
We study the stability problem of unbounded shear flow, with velocity U(i)=Sx(3)delta(i1), subjected to a uniform vertical density stratification, with Brunt-Väisälä frequency N, and system rotation of rate Omega about an axis aligned with the spanwise (x(2)) direction. The evolution of plane-wave disturbances in this shear flow is governed by a nonhomogeneous second-order differential equation with time-dependent coefficients. An analytical solution is found to be described by Legendre functions in terms of the nondimensional parameter sigma(phi)(2)=R(R+1)sin(2) phi+R(i), where R=(2Omega/S) is the rotation number, phi is the angle between the horizontal wave vector and the streamwise axis, and R(i)=N(2)/S(2) is the Richardson number. The long-time behavior of the solution is analyzed using the asymptotic representations of the Legendre functions. On the one hand, linear stability is analyzed in terms of exponential growth, as in a normal-mode analysis: the rotating stratified shear flow is stable if R(i)>1/4, or if 0
We consider unbounded precessing rotating flows in which vertical or horizontal shear is induced by the interaction between the solid-body rotation (with angular velocity Omega(0)) and the additional "precessing" Coriolis force (with angular velocity -epsilonOmega(0)), normal to it. A "weak" shear flow, with rate 2epsilon of the same order of the Poincaré "small" ratio epsilon , is needed for balancing the gyroscopic torque, so that the whole flow satisfies Euler's equations in the precessing frame (the so-called admissibility conditions). The base flow case with vertical shear (its cross-gradient direction is aligned with the main angular velocity) corresponds to Mahalov's [Phys. Fluids A 5, 891 (1993)] precessing infinite cylinder base flow (ignoring boundary conditions), while the base flow case with horizontal shear (its cross-gradient direction is normal to both main and precessing angular velocities) corresponds to the unbounded precessing rotating shear flow considered by Kerswell [Geophys. Astrophys. Fluid Dyn. 72, 107 (1993)]. We show that both these base flows satisfy the admissibility conditions and can support disturbances in terms of advected Fourier modes. Because the admissibility conditions cannot select one case with respect to the other, a more physical derivation is sought: Both flows are deduced from Poincaré's [Bull. Astron. 27, 321 (1910)] basic state of a precessing spheroidal container, in the limit of small epsilon . A Rapid distortion theory (RDT) type of stability analysis is then performed for the previously mentioned disturbances, for both base flows. The stability analysis of the Kerswell base flow, using Floquet's theory, is recovered, and its counterpart for the Mahalov base flow is presented. Typical growth rates are found to be the same for both flows at very small epsilon , but significant differences are obtained regarding growth rates and widths of instability bands, if larger epsilon values, up to 0.2, are considered. Finally, both flow cases are briefly discussed in view of a subsequent nonlinear study using pseudospectral direct numerical simulations, which is a natural continuation of RDT.
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