Time-step coupling for hybrid simulations of multiscale flows

A new method is presented for the exploitation of time-scale separation in hybrid continuum-molecular models of multiscale flows. Our method is a generalisation of existing approaches, and is evaluated in terms of computational efficiency and physical/numerical error. Comparison with existing schemes demonstrates comparable, or much improved, physical accuracy, at comparable, or far greater, efficiency (in terms of the number of time-step operations required to cover the same physical time). A leapfrog coupling is proposed between the 'macro' and 'micro' components of the hybrid model and demonstrates potential for improved numerical accuracy over a standard simultaneous approach. A general algorithm for a coupled time step is presented. Three test cases are considered where the degree of time-scale separation naturally varies during the course of the simulation. First, the step response of a second-order system composed of two linearly-coupled ODEs. Second, a micro-jet actuator combining a kinetic treatment in a small flow region where rarefaction is important with a simple ODE enforcing mass conservation in a much larger spatial region. Finally, the transient start-up flow of a journal bearing with a cylindrical rarefied gas layer. Our new time-stepping method consistently demonstrates as good as or better performance than existing schemes. This superior overall performance is due to an adaptability inherent in the method, which allows the most-desirable aspects of existing schemes to be applied only in the appropriate conditions.