A simple stabilized finite element method for solving two phase compressible–incompressible interface flows

Abstract When computing interface flows between compressible (gas) and incompressible (liquid) fluids, one faces at least to the following difficulties: (1) transition from a gas density linked to the local temperature and pressure by an equation of state to a liquid density mainly constant in space, (2) proper approximation of the divergence constraint in incompressible regions and (3) wave transmission at the interface. The aim of the present paper is to design a global (i.e. the same for each phase) numerical method to address easily this coupling. To this end, the same set of primitive unknowns and equations is used everywhere in the flow, but with a dynamic parameterization that changes from compressible to incompressible regions. On one hand, the compressible Navier–Stokes equations are considered under weakly compressibility assumption so that a non-conservative formulation can be used. On the other hand, the incompressible non-isothermal model is retained. In addition, the level set transport equation is used to capture the interface position needed to identify the local characteristics of the fluid and to recover the adequate local modelling. For space approximation of Navier–Stokes equations, a Galerkin least-squares finite element method is used. Two essential elements for defining this numerical scheme are the stabilization and the computation of element integral of the approximated weak form. Since very different concerns motivate the need for stabilization in compressible and incompressible flows, the first difficulty is to design a stabilization operator suitable for both types of flows especially in mixed elements. In addition, some integral of discontinuous functions must be correctly computed to ensure interfacial wave transmission. To overcome these two difficulties, specific averages are computed especially near the interface. Finally, the level set transport equation is computed by a quadrature free Discontinuous Galerkin method. Numerical strategies are performed and validated for 1D and 2D applications.

[1]  C. W. Hirt,et al.  Volume of fluid (VOF) method for the dynamics of free boundaries , 1981 .

[2]  Sandra Tancogne.  Calcul numérique et stabilité d'écoulements diphasiques tridimensionnels en microfluidique , 2007 .

[3]  M. Billaud.  Eléments finis stabilisés pour des écoulements diphasiques compressible-incompressible , 2009 .

[4]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems , 1986 .

[5]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[6]  Bernhard Müller.  Low mach number asymptotics of the navier-stokes equations and numerical implications , 1999 .

[7]  R. Fedkiw,et al.  A numerical method for two-phase flow consisting of separate compressible and incompressible regions , 2000 .

[8]  A. Smolianski.  Finite‐element/level‐set/operator‐splitting (FELSOS) approach for computing two‐fluid unsteady flows with free moving interfaces , 2005 .

[9]  Takashi Yabe,et al.  Unified Numerical Procedure for Compressible and Incompressible Fluid , 1991 .

[10]  Jean-François Remacle,et al.  A quadrature-free discontinuous Galerkin method for the level set equation , 2006, J. Comput. Phys..

[11]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[12]  Nicola Parolini,et al.  Mass preserving finite element implementations of the level set method , 2006 .

[13]  Jean-François Remacle,et al.  A stabilized finite element method using a discontinuous level set approach for solving two phase incompressible flows , 2006, J. Comput. Phys..

[14]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[15]  Guillermo Hauke,et al.  Simple stabilizing matrices for the computation of compressible flows in primitive variables , 2001 .

[16]  T. Alazard,et al.  Low Mach Number Limit of the Full Navier-Stokes Equations , 2005, math/0501386.

[17]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[18]  L. Pesch,et al.  Construction of stabilization operators for Galerkin least-squares discretizations of compressible and incompressible flows , 2007 .

[19]  L. Franca,et al.  Stabilized finite element methods. II: The incompressible Navier-Stokes equations , 1992 .

[20]  Damien Guégan.  Modélisation numérique d'écoulements bifluides 3 D instationnaires avec adaptation de maillage , 2007 .

[21]  Charbel Farhat,et al.  A higher-order generalized ghost fluid method for the poor for the three-dimensional two-phase flow computation of underwater implosions , 2008, J. Comput. Phys..

[22]  H. Guillard,et al.  On the behaviour of upwind schemes in the low Mach number limit , 1999 .

[23]  Feng Xiao,et al.  Unified formulation for compressible and incompressible flows by using multi-integrated moments I: one-dimensional inviscid compressible flow , 2004 .

[24]  Masato Ida.  An improved unified solver for compressible and incompressible fluids involving free surfaces. Part I. Convection , 2000 .

[25]  Martin E. Weber,et al.  Bubbles in viscous liquids: shapes, wakes and velocities , 1981, Journal of Fluid Mechanics.

[26]  Masato Ida.  An improved unified solver for compressible and incompressible fluids involving free surfaces. II. Multi-time-step integration and applications , 2002 .

[27]  Takashi Yabe,et al.  A universal cubic interpolation solver for compressible and incompressible fluids , 1991 .

[28]  Analysis of stabilization operators for Galerkin least-squares discretizations of the incompressible Navier-Stokes equations , 2006 .

[29]  Stanley Osher,et al.  A second order primitive preconditioner for solving all speed multi-phase flows , 2005 .

[30]  Feng Xiao,et al.  Unified formulation for compressible and incompressible flows by using multi-integrated moments II: Multi-dimensional version for compressible and incompressible flows , 2006, J. Comput. Phys..

[31]  Thomas J. R. Hughes,et al.  A comparative study of different sets of variables for solving compressible and incompressible flows , 1998 .

[32]  Ronald Fedkiw,et al.  A method for avoiding the acoustic time step restriction in compressible flow , 2009, J. Comput. Phys..

[33]  M. Baer,et al.  A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials , 1986 .

[34]  R. Abgrall,et al.  A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows , 1999 .

[35]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics. X - The compressible Euler and Navier-Stokes equations , 1991 .

[36]  Jaap J. W. van der Vegt,et al.  A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids , 2008, J. Comput. Phys..

[37]  Guillermo Hauke,et al.  a Unified Approach to Compressible and Incompressible Flows and a New Entropy-Consistent Formulation of the K - Model. , 1994 .

[38]  S. Schochet,et al.  The Incompressible Limit of the Non-Isentropic Euler Equations , 2001 .

[39]  Keh-Ming Shyue,et al.  A fluid-mixture type algorithm for compressible multicomponent flow with Mie-Grüneisen equation of state , 2001 .