Comput Mech DOI 10.1007/s00466-012-0746-2 ORIGINAL PAPER Simulation of single mode Rayleigh–Taylor instability by SPH method M. S. Shadloo · A. Zainali · M. Yildiz Received: 31 December 2011 / Accepted: 13 June 2012 © Springer-Verlag 2012 Abstract A smoothed particle hydrodynamics (SPH) solu- tion to the Rayleigh–Taylor instability (RTI) problem in an incompressible viscous two-phase immiscible fluid with sur- face tension is presented. The present model is validated by solving Laplace’s law, and square bubble deformation with- out surface tension whereby it is shown that the implemented SPH discretization does not produce any artificial surface tension. To further validate the numerical model for the RTI problem, results are quantitatively compared with analyti- cal solutions in a linear regime. It is found that the SPH method slightly overestimates the border of instability. The long time evolution of simulations is presented for investi- gating changes in the topology of rising bubbles and falling spike in RTI, and the computed Froude numbers are com- pared with previous works. It is shown that the numerical algorithm used in this work is capable of capturing the inter- face evolution and growth rate in RTI accurately. Keywords Smoothed particle hydrodynamics (SPH) · Mesh free method · Projection method · Multi-phase flow · Interfacial flow · Rayleigh–Taylor instability (RTI) 1 Introduction Instability developing and evolving at the interface between two horizontal parallel fluids of different viscosities and den- sities with the heavier fluid at the top and the lighter one at the M. S. Shadloo · A. Zainali · M. Yildiz (B ) Faculty of Engineering and Natural Sciences, Advanced Composites and Polymer Processing Laboratory, Sabanci University, Tuzla, 34956 Istanbul, Turkey e-mail: meyildiz@sabanciuniv.edu M. S. Shadloo e-mail: mostafa@sabanciuniv.edu bottom is known as the Rayleigh–Taylor instability (RTI) to honor the pioneering works of Lord Rayleigh [1] and Taylor [2]. The instability initiates when a multiphase fluid system with different densities experiences gravitational force. As a result, an unstable disturbance tends to grow in the direc- tion of gravitational field thereby releasing and reducing the potential energy of the system. Due to being an important phenomenon in many fields of engineering and sciences, the RTI has been widely investi- gated by using experimental [3, 4], analytical [5, 6] as well as numerical [7, 8] approaches. In the literature, one may find many qualitative mesh-dependent numerical studies for this two-phase flow problem [9–16]. There are also a few works that have used the smoothed particle hydrodynamics (SPH) method to model the RTI problem [17–21]. Cummins and Rudman [17] solved an RTI problem using the incom- pressible SPH (ISPH) approach which is based on the pro- jection method. Tartakovsky and Meakin [18] modeled RTI problem in a multiphase and multi-component mixture with the weakly compressible SPH (WCSPH) method through solving momentum and species mass balance equations con- currently. Hu and Adams [19] used the combination of pro- jection methods proposed and implemented by Cummins and Rudman [17] and Shao and Lo [20] and solved the RTI as a benchmark problem. More recently, Grenier et al. [21] pre- sented a new WCSPH formulation for simulating interfa- cial flows, and modeled the RTI to validate their numerical scheme. All of these SPH works and some others simulated the RTI problem as a validation test case for the numerical algorithms. Surprisingly, out of the works which have been published up to now, there are only a few studies, especially for the long time evolution of the RTI, where the authors compare their numerical results with available analytical the- ories and if it is so, mesh dependent techniques were used [10, 16], and to our best knowledge, there is no SPH work to 123