Noname manuscript No. (will be inserted by the editor) Solution of Navier Stokes Equations on a Fixed Mesh using Conforming Enrichment of Velocity and Pressure D. C. Tanyildiz · J. Marti · R. Rossi Received: date / Accepted: date Abstract Simulation of fluid flows of multi-materials is an intriguing topic in Computational Mechanics. Cap- turing the physics of the interface between different materials poses a challenge because of the discontinu- ities that may occur on the interface. Several meth- ods have been proposed in the literature to deal with this issue. In this paper, a technique based on Nitsche’s Method has been employed on a fixed mesh combined with the PFEM-2 strategy for the solution of Navier- Stokes Equations on multi-fluid flows. The novelty of this technique is its capability of capturing the strong and weak discontinuities, and its compatibility for the application of various types of boundary conditions on the interface. Keywords Lagrangian particles · Multi-fluids · Nitsche’s Method · Fixed mesh 1 Introduction Multi-fluid flow is a very common phenomena in the na- ture and therefore its numerical simulation is of great importance. However, this is a challenging task. The first challenge of the simulation of such flows is related with the capturing of the real behaviour of the flow D. C. Tanyildiz Centre Internacional de M` etodes Num` erics en Enginyeria (CIMNE) Gran Capit´an s/n, 08034 Barcelona, Spain E-mail: dcagri@cimne.upc.edu J. Marti CIMNE E-mail: julio.marti@cimne.upc.edu R. Rossi CIMNE E-mail: rrossi@cimne.upc.edu across the interface. This challenge originates from the fact that fluids with different properties may have sharp behaviour changes on the interface. An example for this could be a change in the pressure gradient, or a jump in the velocity field. Apart from the difficulties of cap- turing these physical changes, another challenge is the difficulty of detecting the evolution of the position of the interface throughout the simulation. To circumvent these issues, special care on the interface has to be taken at each time step. Several strategies have been developed for the solu- tion of multi-fluid problems. Eulerian based methodolo- gies use special techniques to capture the interface such as: Level Set approaches[1,2], Volume of Fluid meth- ods[3], etc. Since the element could be cut by the inter- face in any arbitrary position, further techniques, such as X-FEM and E-FEM, are needed for the representa- tion of weak and/or strong discontinuities in the pres- sure and/or velocity fields [5,17,20,25,24,26]. These methodologies exhibit deficiencies in conservation of the mass. Recent studies have introduced novel techniques for alleviating this issue [6–9, 4]. A simple remedy to such difficulties is to employ a mesh based Lagrangian approach, and to carry out the simulation by moving the nodes on the interface and changing the mesh topology. [34,36,10–12]. Unfor- tunately, since the interface is subject to changes, one needs to repeat this procedure in every time-step, lead- ing to a high computational cost. Apart from that, even employing this technique, the nodes at the interface need to be duplicated in order to capture the strong discontinuities across the interface[11]. Moreover, one needs to employ small time steps with mesh based La- grangian methods in order to avoid the element inver- sion throughout the simulation.