ИСТИНА

We consider the stability of a dusty-gas flow in the boundary layer on a flat plate. The flow is described in the framework of the two-fluid model1,2 with an incompressible Newtonian carrier fluid and non-Brownian particles. The particle volume fraction is negligibly small but their mass loading is finite and therefore the two-way coupling model is used. In the interphase momentum exchange, the Stokes drag and the Saffman lift force3 arising due to a large fluid velocity gradient on the particle scale are taken into account. The flow is considered in the region remote from the leading edge of the plate, where the velocities of both phases coincide and correspond to the Blasius profile. The particle number density N(y) is assumed to be nonuniform and have a Gaussian distribution with given thickness () and the coordinate of the maximum (). The set of governing parameters includes also the Reynolds number (Re), the particle inertia parameter or inverse Stokes number (), the parameter determining the lift force scale (K), and the particle mass loading . A parametric study of the first normal mode (modal stability) is performed and the transient growth is investigated, based on the analysis of ‘optimal disturbances’ (non-modal stability). The increment of the most unstable 2D mode is calculated using an orthonormalization method, while the parameters of 3D optimal perturbations are calculated using a finite-difference method and the QR-algorithm. The most pronounced effect of particles on the first mode is indicated in the case when the particles are concentrated in the vicinity of the so-called ‘critical layer’, located at a point where the main-flow velocity is equal to the phase velocity of the mode. In contrast to the pure-fluid flow, in a certain range of governing parameters corresponding to sufficiently small  and  (Fig. 1a), there exist two unstable normal modes. In the case of a sufficiently thick particle layer located close to the plate, the critical Reynolds number increases by two orders of magnitude as compared to the pure- fluid flow. It is found from the calculations that the optimal disturbances are streaky structures. The transient growth is most pronounced when the distance of the particle concentration maximum to the plate is of the order of the displacement thickness (Fig. 1b). For a fixed particle mass concentration averaged over the boundary layer thickness, the maximum energy of optimal disturbances corresponds to the flow with a narrow particle distribution.