Simulation of the droplet evaporation and combustion and droplet impact on a solid surface using a discontinuous Galerkin scheme
Final Report Abstract
The aim of this project was to develop a Discontinuous Galerkin solver for the simulation of droplet evaporation and combustion. Main building blocks for this project were a solver for low-Mach number flows, a solver for species transport and combustion and a solver for multiphase flows. Due to unexpected difficulties during the development and a lack of time we have not been able to arrive at the point, where the coupling of the subproblems for the simulation of droplet combustion would have been possible. Still, we made substantial progress in the development of the individual building blocks. During the course of this project, we developed a Discontinuous Galerkin solver for steady and unsteady incompressible flows based on the SIMPLE algorithm. The development of the incompressible solver was initially not planned in the project proposal, but built a very helpful basis for the later developed low-Mach number solver as well as the developed solver for reactive flows. To support this project a second DFG project was initiated, which solely focused on Discontinuous Galerkin methods for multiphase flows. The solver for steady and unsteady incompressible flows was extensively tested by simulating various test cases. We investigated the accuracy, performance and stability of the developed solver. We performed spatial convergence studies and could demonstrate the high-accuracy of the method. Applying the mixed-order formulation, we observed convergence rates of k + 1 for the velocity and k for the pressure. By simulating the Orr-Sommerfeld problem we could show the stability of the method. Also for the low-Mach number solver we obtained the expected spatial convergence rates of the Discontinuous Galerkin Method. Applying a mixed-order formulation, for Couette flow with a vertical temperature gradient the convergence rates are approximately k + 1 for the velocity and the temperature and k for the pressure. For the common benchmark problem of natural convection in a square cavity we obtained very good agreement with the numerical reference solutions. We could also demonstrate the benefits of using higher-order methods compared to lower-order methods in terms of degrees of freedom and accuracy. The solver was also successfully tested for the case of unsteady natural convection in a tall cavity. The transport equations for the species including the source terms of chemical reactions have been implemented and were coupled to the low-Mach number equations for the flow field. A manufactured solution was constructed to perform h-convergence studies for this rather complex set of equations. Also for this test case including chemical reactions we observed the expected convergence rates. Within the second project the multiphase solver was pushed forward. Filter algorithms for the calculation of the interface curvature were developed, which are important for an accurate surface tension. As a test case the temporal evolution of an initially distorted droplet subjected to a quiescent flow field was simulated.
Publications
- (2016). A high-order Discontinuous Galerkin solver for low-Mach number flows. International Journal for Numerical Methods in Fluids
Klein, B., Müller, B., Kummer, F. and Oberlack, M.
(See online at https://doi.org/10.1002/fld.4193) - (2013). A SIMPLE based discontinuous Galerkin solver for steady incompressible flows. Journal of Computational Physics, 237, pp.235–250
Klein, B., Kummer, F. and Oberlack, M.
(See online at https://doi.org/10.1016/j.jcp.2012.11.051) - (2015). A high-order Discontinuous Galerkin solver for incompressible and lowa Mach number flows. Ph.D. thesis. Technische Universität Darmstadt
Klein, B.
- (2015). An extension of the SIMPLE based discontinuous Galerkin solver to unsteady incompressible flows. International Journal for Numerical Methods in Fluids, 77(10), pp.571–589
Klein, B., Kummer, F., Keil, M. and Oberlack, M.
(See online at https://doi.org/10.1002/fld.3994)
