Multi-fluid dynamical model of isothermal gas and buoyant dispersed particles: Monodisperse mixture, reference solution of DustyWave problem as test for CFD-solvers, effective sound speed for high and low mutual drag

A computer simulation of the dynamics of dense mixtures of gas and solid particles based on a macroscopic approach is used in studying technological devices and natural phenomena. A numerical solution of equations constituting a mathematical model of medium dynamics runs into difficulties. In particular, the problems, in which characteristic time of mass, momentum or energy exchange between phases is much less than dynamical time, are stiff and thus complicated. In this case, explicit methods of integration appear to require a small timestep. On the other hand, implicit methods provide reasonable computational costs for such stiff problems, but some of them are known to suffer from huge approximation error. In this work, we propose a test problem that allows evaluating the accuracy of interphase momentum exchange computing for any intensity including computationally expensive regimes. The test problem is a problem of propagation of sound waves in the mixture of isothermal gas and buoyant monodisperse particles. The formulation of the problem is based on a two-fluid approach; motion equations include terms involving momentum exchange between gas and dispersed particles and the buoyancy of dispersed phase. The problem is solved analytically by using the Fourier method and a dispersion analysis. In the general case, the solution is generated numerically for an arbitrary value of relaxation time by using the downloadable code that we developed. In particular cases (such as infinitely short time of velocity relaxation or relaxation equilibrium and infinitely long time of velocity relaxation or frozen equilibrium), we determined efficient sound speed in a gas–dust medium and thus obtained simple analytical representations of the solution. We showed that the obtained analytical solutions of the problem are reproduced by an Euler and Lagrangian numerical method, and thus can be recommended as reference solution for numerical codes benchmarking.