LINEAR-STABILITY ANALYSIS OF PLANE QUADRATIC FLOWS IN A ROTATING-FRAME WITH APPLICATIONS TO MODELING

The linear response of turbulence to a distortion and simple rotation is investigated in this paper from a fundamental theoretical standpoin t. Quadratic flows are a special case of planar flows with constant me an velocity gradients, which can be characterized by a constant rate o f strain D and a constant spanwise (normal to the plane) vorticity com ponent -W=2 Omega(0), with arbitrary values (here, we can take D>0 and W>0 without any loss of generality). According to the sign of D-2 -Om ega(0)(2), streamlines are hyperbolic, rectilinear (pure shear flow) o r elliptic. Since these flows can also be considered as mean flows, wh en superimposing a three-dimensional disturbance (or fluctuating turbu lent) field which satisfies statistical homogeneit), the linearized an alysis of the disturbance field is of interest both from the point of view of hydrodynamic stability (e.g., the elliptical flow instability) and from the point of view of homogeneous rapid distortion theory (RD T) including applications to the basic statistics. The case of quadrat ic flow in a rotating frame (with angular velocity Omega in the direct ion normal to the plane of the basic flow) is revisited in this paper, in order to complete-with the three parameters D, -W=2 Omega(0) and O mega-previous works on linear theory by Cambon et al. [J. Fluid Mech. 278, 175 (1994)] and Speziale, Abid, and Blaisdell [Phys. Fluids 8, 78 1 (1996)]. From a simplified ''pressure-less'' linear approach, a gene ral stability criterion is derived based on the value of the modified Bradshaw number B-r=[D-2-(2 Omega+Omega(0))(2)]/S-2 (S=D-Omega(0)), wh ich coincides with the rotational Richardson number introduced by Brad shaw [J. Fluid Mech. 36, 177 (1969)] and denoted by B=2 Omega-(S- 2 Om ega)/S-2 (in particular, for the case of pure shear flow where D=W/2=- Omega(0)=S/2). It is shown that this criterion gives results identical to the ''true'' linear stability analysis (including the effect of th e fluctuating pressure) if the absolute vorticity 2 Omega+2 Omega(0) h as a zero value. In addition, the relevance of this criterion is check ed with respect to the true linear approach in distorted wave space an d related RDT applications. For all of the cases, the maximum amplific ation for the three-dimensional disturbance field is found for zero ti lting vorticity 2 Omega+Omega(0) and for pure spanwise modes (with wav e vector normal to the plane of the quadratic flow), in accordance wit h the generalized Bradshaw criterion and other results in hydrodynamic stability. For other spectral directions, the agreement is not as com plete except for the pure shear case, and this is particularly discuss ed looking at statistical RDT solutions, which involve a summation ove r all directions of the wave vector. Finally, the impact of the whole analysis on second-order, one-point modeling is discussed. (C) 1997 Am erican Institute of Physics.