The theorem of the complement for nested subpfaffian sets

Let R be an o-minimal expansion of the real field, and let L(R) be the language consisting of all nested Rolle leaves over R. We call a set nested subpfaffian over R if it is the projection of a boolean combination of definable sets and nested Rolle leaves over R. Assuming that R admits analytic cell decomposition, we prove that the complement of a nested subpfaffian set over R is again a nested subpfaffian set over R. As a corollary, we obtain that if R admits analytic cell decomposition, then the pfaffian closure P(R) of R is obtained by adding to R all nested Rolle leaves over R, a one-stage process, and that P(R) is model complete in the language L(R).Comment: final version before publicatio

New results on the model problem of the diffusion of turbulence from a plane source

The problem of the diffusion of turbulence from a plane source is addressed in the context of two-equation eddy-viscosity models and Reynolds-stress-transport models. In the steady state, full analytic solutions are given. At second order, they provide the equilibrium value of the anisotropy level obtained with different combinations of return-to-isotropy and turbulent-diffusion schemes and confirm the results obtained by Straatman et al. [AIAA J. 36, 929 (1998)] in an approximate analysis. In addition, all the characteristics of the turbulence decrease can be determined and it is shown that a special constraint on the value of the modeling constants should hold if turbulence fills the whole surrounding space. In a second step, precise results can be given for the unsteady model problem at the first-order-closure level. The evolution cannot be described with a single set of characteristic scales and one has to distinguish the cases of short and large times. In the short-time regime, the flow is governed by the characteristic scales of turbulence at the source and contamination of the flow proceeds as t^1/2. At large times, the flow is governed by time-dependent characteristic scales that correspond to the solution of the steady problem at the instantaneous location of the front. Contamination of the flow proceeds as a power of time that can be related to the value of the modeling constants. The role of a combination of these constants is emphasized whose value can be specified to produce a solution that matches simultaneously the experimental data for the decrease of turbulent kinetic energy in the steady state and the exponent of the propagation velocity in the transient regime

The structure of the solution obtained with Reynolds-stress-transport models at the free-stream edges of turbulent flows

The behavior of Reynolds-stress-transport models at the free-stream edges of turbulent flows is investigated. Current turbulent-diffusion models are found to produce propagative (possibly weak) solutions of the same type as those reported earlier by Cazalbou, Spalart, and Bradshaw [Phys. Fluids 6, 1797 (1994)] for two-equation models. As in the latter study, an analysis is presented that provides qualitative information on the flow structure predicted near the edge if a condition on the values of the diffusion constants is satisfied. In this case, the solution appears to be fairly insensitive to the residual free-stream turbulence levels needed with conventional numerical methods. The main specific result is that, depending on the diffusion model, the propagative solution can force turbulence toward definite and rather extreme anisotropy states at the edge (one - or two-component limit). This is not the case with the model of Daly and Harlow [Phys. Fluids 13, 2634 (1970)]; it may be one of the reasons why this "old" scheme is still the most widely used, even in recent Reynolds-stress-transport models. In addition, the analysis helps us to interpret some difficulties encountered in computing even very simple flows with Lumley's pressure-diffusion model [Adv. Appl. Mech. 18, 123 (1978)]. A new realizability condition, according to which the diffusion model should not globally become "anti-diffusive", is introduced, and a recalibration of Lumley's model satisfying this condition is performed using information drawn from the analysis