A space-time discontinuous Galerkin finite element method for two-fluid problems

A space-time discontinuous Galerkin finite element method for two fluid flow problems is presented. By using a combination of level set and cut-cell methods the interface between two fluids is tracked in space-time. The movement of the interface in space-time is calculated by solving the level set equation, where the interface geometry is identified with the 0-level set. To enhance the accuracy of the interface approximation the level set function is advected with the interface velocity, which for this purpose is extended into the domain. Close to the interface the mesh is locally refined in such a way that the 0-level set coincides with a set of faces in the mesh. The two fluid flow equations are solved on this refined mesh. The procedure is repeated until both the mesh and the flow solution have converged to a reasonable accuracy.\ud The method is tested on linear advection and Euler shock tube problems involving ideal gas and compressible bubbly magma. Oscillations around the interface are eliminated by choosing a suitable interface flux

A Volume of Fluid (VoF) based all-mach HLLC Solver for Multi-Phase Compressible Flow with Surface-Tension

This work presents an all-Mach method for two-phase inviscid flow in the presence of surface tension. A modified version of the Hartens, Lax, Leer and Contact (HLLC) approximate Riemann solver based on Garrick et al. [1] is developed and combined with the popular Volume of Fluid (VoF) method: Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM). This novel combination yields a scheme with both HLLC shock capturing as well as accurate liquid-gas interface tracking characteristics. To ensure compatibility with VoF, the Monotone Upstream-centred Scheme for Conservation Laws (MUSCL) [2] is applied to non-conservative (primitive) variables, which yields both robustness and accuracy. Liquid-gas interface curvature is computed via both height functions [3, 4] and the convolution method [5]. This is in the interest of applicability to both cartesian and arbitrary meshes. The author emphasizes the use of VoF in the interest of surface tension modelling accuracy. The method is validated using a range of test-cases available in literature. The results show flow features that are in agreement with experimental and benchmark data. In particular, the use of the HLLC-VoF combination leads to a sharp volume fraction and energy field with improved accuracy (up to secondorder)

MFC: An open-source high-order multi-component, multi-phase, and multi-scale compressible flow solver

MFC is an open-source tool for solving multi-component, multi-phase, and bubbly compressible flows. It is capable of efficiently solving a wide range of flows, including droplet atomization, shock–bubble interaction, and bubble dynamics. We present the 5- and 6-equation thermodynamically-consistent diffuse-interface models we use to handle such flows, which are coupled to high-order interface-capturing methods, HLL-type Riemann solvers, and TVD time-integration schemes that are capable of simulating unsteady flows with strong shocks. The numerical methods are implemented in a flexible, modular framework that is amenable to future development. The methods we employ are validated via comparisons to experimental results for shock–bubble, shock–droplet, and shock–water-cylinder interaction problems and verified to be free of spurious oscillations for material-interface advection and gas–liquid Riemann problems. For smooth solutions, such as the advection of an isentropic vortex, the methods are verified to be high-order accurate. Illustrative examples involving shock–bubble-vessel-wall and acoustic–bubble-net interactions are used to demonstrate the full capabilities of MFC

Direct numerical simulation of compressible interfacial multiphase flows using a mass-momentum-energy consistent volume-of-fluid method

Compressible interfacial multiphase flows (CIMF) are essential to different applications, such as liquid fuel injection in supersonic propulsion systems. Since high-level details in CIMF are often difficult to measure in experiments, numerical simulation is an important alternative to shed light on the unclear physics. A direct numerical simulation (DNS) of CIMF will need to rigorously resolve the shock waves, the interfaces, and the interaction between the two. A novel numerical method has been developed and implemented in the present study. The geometric volume-of-fluid (VOF) method is employed to resolve the sharp interfaces between the two phases. The advection of the density, momentum, and energy is carried out consistently with VOF advection. To suppress spurious oscillations near shocks, numerical diffusion is introduced based on the Kurganov-Tadmor method in the region away from the interface. The contribution of pressure is incorporated using the projection method and the pressure is obtained by solving the Poisson-Helmholtz equation, which allows the present method to handle flows with all Mach numbers. The present method is tested by a sequence of CIMF problems. The simulation results are validated against theories, experiments, and other simulations, and excellent agreement has been achieved. In particular, the linear single-mode Richtmyer-Meshkov instabilities with finite Weber and Reynolds numbers are simulated. The simulation results agree very well with the linear stability theory, which affirms the capability of the present method in capturing the viscous and capillary effects on shock-interface interaction

Modified HLLC-VOF solver for incompressible two-phase fluid flows

A modified HLLC-type contact preserving Riemann solver for incompressible two-phase flows using the artificial compressibility formulation is presented. Here, the density is omitted from the pressure evolution equation. Also, while calculating the eigenvalues and eigenvectors, the variations of the volume fraction is taken into account. Hence, the equations for the intermediate states and the intermediate wave speed are different from the previous HLLC-VOF formulation [Bhat S P and Mandal J C, J. Comput. Phys. 379 (2019), pp. 173-191]. Additionally, an interface compression algorithm is used in tandem to ensure sharp interfaces. The modified Riemann solver is found to be robust compared to the previous HLLC-VOF solver, and the results produced are superior compared to non-contact preserving solver. Several test problems in two- and three-dimensions are solved to evaluate the efficacy of the solver on structured and unstructured meshes

Finite-volume WENO scheme for viscous compressible multicomponent flows

We develop a shock- and interface-capturing numerical method that is suitable for the simulation of multicomponent flows governed by the compressible Navier–Stokes equations. The numerical method is high-order accurate in smooth regions of the flow, discretely conserves the mass of each component, as well as the total momentum and energy, and is oscillation-free, i.e. it does not introduce spurious oscillations at the locations of shockwaves and/or material interfaces. The method is of Godunov-type and utilizes a fifth-order, finite-volume, weighted essentially non-oscillatory (WENO) scheme for the spatial reconstruction and a Harten–Lax–van Leer contact (HLLC) approximate Riemann solver to upwind the fluxes. A third-order total variation diminishing (TVD) Runge–Kutta (RK) algorithm is employed to march the solution in time. The derivation is generalized to three dimensions and nonuniform Cartesian grids. A two-point, fourth-order, Gaussian quadrature rule is utilized to build the spatial averages of the reconstructed variables inside the cells, as well as at cell boundaries. The algorithm is therefore fourth-order accurate in space and third-order accurate in time in smooth regions of the flow. We corroborate the properties of our numerical method by considering several challenging one-, two- and three-dimensional test cases, the most complex of which is the asymmetric collapse of an air bubble submerged in a cylindrical water cavity that is embedded in 10% gelatin

A time splitting projection scheme for compressible two-phase flows. Application to the interaction of bubbles with ultrasound waves

This paper is focused on the numerical simulation of the interaction of an ultrasound wave with a bubble. Our interest is to develop a fully compressible solver in the two phases and to account for surface tension effects. As the volume oscillation of the bubble occurs in a low Mach number regime, a specific care must be paid to the effectiveness of the numerical method which is chosen to solve the compressible Euler equations. Three different numerical solvers, an explicit HLLC (Harten–Lax–van Leer-Contact) solver [48], a preconditioning explicit HLLC solver [14] and the compressible projection method , and , are described and assessed with a one dimensional spherical benchmark. From this preliminary test, we can conclude that the compressible projection method outclasses the other two, whether the spatial accuracy or the time step stability are considered. Multidimensional numerical simulations are next performed. As a basic implementation of the surface tension leads to strong spurious currents and numerical instabilities, a specific velocity/pressure time splitting is proposed to overcome this issue. Numerical evidences of the efficiency of this new numerical scheme are provided, since both the accuracy and the stability of the overall algorithm are enhanced if this new time splitting is used. Finally, the numerical simulation of the interaction of a moving and deformable bubble with a plane wave is presented in order to bring out the ability of the new method in a more complex situation