Full Lagrangian methods for calculating particle concentration fields in dilute gas-particle flows

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The paper discusses two Full Lagrangian methods for calculating the particle concentration and velocity fields in dilute gas-particle flows. The methods are mathematically similar but crucially different in application. By examining the analytical solution for two-phase stagnation-point flow, it is shown that Osiptsov's method is more general than that of Fernandez de la Mora and Rosner. In Osiptsov's method, the Jacobian of the Eulerian–Lagrangian transformation is computed by integration along particle pathlines. The particle concentration is then obtained algebraically from the Lagrangian form of the particle continuity equation. It is shown that the correct specification of the initial conditions is non-trivial and of vital importance. A technique to alleviate problems of mathematical ‘stiffness’ at small Stokes numbers is also described. Full Lagrangian methods require knowledge of the fluid velocity gradient field and, if the carrier flowfield is calculated numerically, differentiation of a ‘noisy’ field can result in serious errors. The paper describes a method for reducing these errors. The incompressible, inviscid flow over a cylinder provides a useful test case for validation and the Osiptsov method proves its worth by revealing a region, hitherto unknown, of crossing particle pathlines in the mathematical solution. Crossing pathlines and their relationship to Robinson's integral are then discussed, and calculations of particle flow through a turbine cascade at high Mach numbers are presented to illustrate the engineering potential of the method.