Lagrangian Space-Time Methods for Multi-Fluid Problems on Unstructured Meshes (STiMulUs)

**Principal Investigator:**

Walter Boscheri

**Affiliation:**

Department of Civil, Environmental and Mechanical Engineering, University of Trento (Italy)

**Local Project ID:**

stimulus

**HPC Platform used:**

SuperMUC of LRZ

**Date published:**

Many real world processes are modelled using time-dependent partial differential equations (PDE), which are based on the conservation of some physical quantities, like mass or total energy. Therefore these mathematical and physical models are typically addressed as *conservation laws* and they cover a wide range of phenomena, such as environmental and meteorological flows, hydrodynamic and thermodynamic problems, plasma flows as well as the dynamics of many industrial and mechanical processes. In this context fluid dynamics is also governed by such conservation laws, which can be formulated using either an Eulerian approach, where the fluid motion is looked and analyzed from a fixed location, or a Lagrangian approach, in which the observer moves together with a fluid particle. Such physical models cannot in principle be solved analytically, i.e. with an explicit mathematical formulation, hence leading to the development of numerical methods that aim at approximating the physical solution at the aid of computers. To do so, one needs to discretize the computational domain with a so-called *computational mesh* that in this case is supposed to be unstructured, hence considering triangular and tetrahedral elements in two and three space dimensions. This allows our algorithm to be applied also to complex computational domains which are typically present in real world conditions.

A lot of research has been carried out in the last decades in order to develop Lagrangian numerical schemes, because they lead to the main advantage of tracking and precisely identifying material interfaces and contact waves. Therefore a much better resolution is achieved rather than adopting the classical Eulerian formulation with fix computational grid. Lagrangian schemes have been designed and developed in order to compute the flow variables by moving together with the fluid. As a consequence the computational mesh continuously changes its configuration in time, following as close as possible the flow motion by moving with a mesh velocity that must be carefully assigned.

The challenge of any Lagrangian numerical scheme is to preserve at the same time the excellent properties in the resolution of the fluid flow typically achieved by Lagrangian algorithms together with a good mesh quality without invalid elements. This research focuses on the development of finite volume Lagrangian numerical schemes on multidimensional unstructured meshes for fluid dynamic problems. The numerical algorithms developed in the Stimulus project are designed to be high order accurate in space as well as in time, requiring even more information to be updated and recomputed continuously as the simulation goes on.

Due to the above-mentioned procedures and techniques, Lagrangian schemes are typically very demanding in terms of computational efforts. Therefore one should rely on an efficient MPI parallelization of the entire code in order to take advantage from the computational resources of supercomputers. The researchers have focused their attention to **multi-fluid **problems, such as industrial applications which are concerned with *compressible* multi-phase flows as they appear for example in combustion processes of liquid and solid fuels in car, aircraft and rocket engines, but also in explosion and detonation processes. The scientists consider the Baer-Nunziato model for compressible two-phase flow to simulate such phenomena, which has been introduced in 1986 for detonation waves in solid-gas combustion processes.

**Acknowledgments:**

Project STiMulUs was made possible through PRACE (Partnership for Advanced Computing in Europe) with HPC system SuperMUC of the Leibniz Supercomputing Centre (LRZ) serving as computing platform.

**Scientific Contact:**

Dr. Ing. Walter Boscheri

Department of Civil, Environmental and Mechanical Engineering

University of Trento

via Mesiano, 77, I-38123 Trento (Italy)

e-mail: walter.boscheri [@] unitn.it