Compressible Multi-fluid Flows Research Articles

Saturation of Vortex Rings Ejected from Shock-Accelerated Interfaces.

Structures evoking vortex rings can be discerned in shock-accelerated flows ranging from astrophysics to inertial confinement fusion. By constructing an analogy between vortex rings produced in conventional propulsion systems and rings generated by a shock impinging upon a high-aspect-ratio protrusion along a material interface, we extend classical, constant-density vortex-ring theory to compressible multifluid flows. We further demonstrate saturation of such vortex rings as the protrusion aspect ratio is increased, thus explaining morphological differences observed in practice.

Taming the “stiff stiffness” of pressure work and equilibration in numerical schemes for compressible multi-fluid flows

Designing models and schemes for compressible multi-fluid flow is often considered as challenging when dealing with contrasted equations of state, low volume fractions, or high compression or expansion ratios. This is due to the potentially severe stiffness of pressure couplings, aggravated by (i) their quadratic multiplicity as they connect all energy reservoirs (ii) the entropy conditions to be preserved on each fluid, (iii) their potentially unknown signs, and (iv) their potential “stiff stiffness”, defined in the present work. Whether captured under pressure-equilibration or pressure-relaxation models—where fluids respectively share a common pressure or undergo a damped evolution towards a common pressure—the stiff terms of the evolution equations have always required dedicated numerical approaches, with sometimes mixed results depending on specific applications.Some broad general guidelines are here proposed to tame the stiff stiffness issues of the pressure terms. The framework is that of pressure-equilibrated average-field (or Euler–Euler) models, discretized with a splitting of momentum and internal energy equations: this makes the stiff terms behave locally as simple ODEs, amenable to integration by simple schemes. The present approach resorts to (partly) exponential integrators to provide explicit estimations of implicit pressures. The final explicit scheme, though not as robust as would its fully implicit version, displays a vastly improved resilience to stiffness.Numerical tests were carried out on strenuous versions of usual 1D shock tubes on two-fluid mixtures of air and water (described by ideal and stiffened gases). In view of the excellent results, further 1D tests in more extreme conditions were considered: free expansion to vacuum and shocks on zero pressure state (Noh’s test). At usual acoustic CFL values, expansion was robustly simulated for air volume fractions from less than 10−12 to up to nearly 1. Similarly, shocks on zero pressure states were simulated to different final states for air volume fractions as low as 10−8, with air density factors across shocks ranging between a few hundreds and hundredths depending on the choice of dissipation in each fluid. All tests were carried out on a previously developed Geometry, Energy, and Entropy Consistent (GEEC) scheme (Vazquez-Gonzalez et al., 2020).

A five-equation model based global ale method for compressible multifluid and multiphase flows

In this work, a mixing cell closure model was derived in a Lagrangian formulation based on the hypothesis of isentropic compression or expansion. The proposed closure model is used to construct a five-equation model for the simulation of multifluid and multiphase flows. Thereafter a global ALE(Arbitrary Lagrangian-Eulerian) method was developed for the five-equation model. The five-equation based ALE method is a two-stage ALE method, including a Lagrangian phase and a rezone-remap phase. In the first phase, the model equation was discretized with a cell-centered Lagrangian scheme, and a multifluid Riemann solver was proposed by extending traditional Riemann solver to mixed cells with different materials. The five-equation based ALE method can be used to simulate multi-material flows with large deformation, which is a challenge for many traditional ALE methods. Moreover, it can be used to simulate multiphase flows. A series of multi-material and multiphase test problems were simulated with the five-equation based ALE method, and numerical results agree well with exact solutions and reference results.

A mimetic numerical scheme for multi-fluid flows with thermodynamic and geometric compatibility on an arbitrarily moving grid

Simulating transient and compressible multi-fluid flows in extreme situations such as Inertial Confinement Fusion is especially challenging because of numerous and sometimes conflicting constraints : large number of fluids, both isentropic and strongly shocked compressible evolution, highly variable, stiff or contrasted equations of state, large heat sources, large deformations, and transport over large distances. Models and schemes for such flows all share a common non-dissipative “backbone” structure of per-fluid mass, momentum, and energy evolution-and-transport equations, coupled through pressure terms. A novel and efficient multi-fluid numerical scheme for discretizing the backbone equations over a moving grid (ALE or Arbitrary Lagrangian–Eulerian) is here generated through a “Geometry, Energy, and Entropy Compatible” mimicking procedure [Eur. J. Mech. B – Fluids 67, 494 (2017)]. Starting from the discretized density fields, energy fields, and transport operators, the procedure yields the discrete evolution equations in a practically univocal way. With arbitrarily moving grids, number of fluids, contrasts of volume fractions and equations of state, the resulting scheme is fully conservative in masses, momentum, and energy, preserves isentropic behavior to the scheme order, and ensures per-fluid thermodynamic consistency. Noticeably, optimal isentropic behavior is obtained thanks to a non-standard downwind form of pressure gradient. Multi-fluid numerical test cases—including Sod’s shock tube, Ransom’s faucet, and a nine-fluids crossing test—are performed in two-dimensions using deliberately strenuous grid motion strategies. The results confirm the expected properties and illustrate the robustness, stability and versatility of the scheme at finite resolution, though it is not intended to be used as is. The scheme is a foundation block to be complemented with (system-dependent and ubiquitous) physical dissipation terms which provide the necessary damping of divergingly unstable modes at vanishing wavelengths.

Numerical simulation of compressible multifluid flows using an adaptive positivity‐preserving RKDG‐GFM approach

SummaryThe Runge‐Kutta discontinuous Galerkin method together with a refined real‐ghost fluid method is incorporated into an adaptive mesh refinement environment for solving compressible multifluid flows, where the level set method is used to capture the moving material interface. To ensure that the Riemann problem is exactly along the normal direction of the material interface, a simple and efficient modification is introduced into the original real‐ghost fluid method for constructing the interfacial Riemann problem, and the initial conditions of the Riemann problem are obtained directly from the solution polynomials of the discontinuous Galerkin finite element space. In addition, a positivity‐preserving limiter is introduced into the Runge‐Kutta discontinuous Galerkin method to suppress the failure of preserving positivity of density or pressure for the problems involving strong shock wave or shock interaction with material interface. For interfacial cells in adaptive mesh refinement, the data transfer between different grid levels is achieved by using a L2 projection approach along with the least squares fitting. Various numerical cases, including multifluid shock tubes, underwater explosions, and shock‐induced collapse of a underwater air bubble, are computed to assess the capability of the present adaptive positivity‐preserving RKDG‐GFM approach, and the simulated results show that the present approach is quite robust and can provide relatively reasonable results across a wide variety of flow regimes, even for problems involving strong shock wave or shock wave impacting high acoustic impedance mismatch material interface.

The simulation of compressible multi-fluid flows by a GRP-based energy-splitting method

This paper serves to simulate compressible multi-fluid flows using an energy-splitting method based on the generalized Riemann problem (GRP) solver. The emphasis is put on two-dimensional problems to address the fundamental role of transversal effect in the construction of numerical fluxes, which exhibits the genuine multi-dimensionality of the underlying numerical schemes. In contrast with the popularly-used methods of line which use exact or approximate Riemann solvers, the GRP solver, as the discontinuous version of the Lax–Wendroff method, takes the temporal-spatial coupling process to include the transversal effect. Combined with the energy-splitting technique developed earlier in Lei and Li [24], the resolution of flow fields around material interfaces is improved significantly.

A non-oscillatory energy-splitting method for the computation of compressible multi-fluid flows

This paper proposes a new non-oscillatory energy-splitting conservative algorithm for computing multi-fluid flows in the Eulerian framework. In comparison with existing multi-fluid algorithms in the literature, it is shown that the mass fraction model with isobaric hypothesis is a plausible choice for designing numerical methods for multi-fluid flows. Then we construct a conservative Godunov-based scheme with the high order accurate extension by using the generalized Riemann problem solver, through the detailed analysis of kinetic energy exchange when fluids are mixed under the hypothesis of isobaric equilibrium. Numerical experiments are carried out for the shock-interface interaction and shock-bubble interaction problems, which display the excellent performance of this type of schemes and demonstrate that nonphysical oscillations are suppressed around material interfaces substantially.

A compressive interface-capturing scheme for computation of compressible multi-fluid flows

Abstract In this paper, a compressive high-resolution interface-capturing scheme is presented for the computation of compressible multi-fluid flows with high-density ratios and strong shocks. The proposed scheme is coupled with a preconditioned dual-time compressible mixture solver for robust and accurate computations over a wide range of Mach numbers. The scheme is simple and relatively easy to implement. It does not require any calculations for the interface curvature and the normal vector. The numerical approximations were implemented on general, structured grids using an implicit MUSCL upwind approach. Validation tests were conducted for a single reversible vortex, advection of an air-water interface, dam-break flow, and air shock-helium bubble interaction. Finally, a three-dimensional gas-lift flow is presented to demonstrate the capability of the present scheme for handling an interface with large jumps in pressure, temperature, and density.

A low diffusive Lagrange-remap scheme for the simulation of violent air–water free-surface flows

In 2002, Després and Lagoutière [17] proposed a low-diffusive advection scheme for pure transport equation problems, which is particularly accurate for step-shaped solutions, and thus suited for interface tracking procedure by a color function. This has been extended by Kokh and Lagoutière [28] in the context of compressible multifluid flows using a five-equation model. In this paper, we explore a simplified variant approach for gas–liquid three-equation models. The Eulerian numerical scheme has two ingredients: a robust remapped Lagrange solver for the solution of the volume-averaged equations, and a low diffusive compressive scheme for the advection of the gas mass fraction. Numerical experiments show the performance of the computational approach on various flow reference problems: dam break, sloshing of a tank filled with water, water–water impact and finally a case of Rayleigh–Taylor instability. One of the advantages of the present interface capturing solver is its natural implementation on parallel processors or computers.

A volume of fluid method based ghost fluid method for compressible multi-fluid flows

A ghost fluid method for compressible multi-fluid flows is presented in an adaptive mesh refinement (AMR) environment, where the volume of fluid method is used to track the interface. Various numerical examples are presented to compare the proposed method with interface capturing methods using pressure–temperature equilibrium and non-equilibrium temperature mixed cell approaches. It is found both mixing models are unable to generate accurate results for strong shock refractions through high acoustic impedance mismatch interfaces. The proposed method is found to be quite robust and can provide relatively reasonable results across a wide variety of flow regimes. The ghost fluid coupling between the fluid solver and the volume of fluid method is designed to be simple and consistent in any spatial dimension on AMR grid.

INTERACTION OF SHOCK WAVE WITH MULTI-FLUIDS INTERFACE USING QUADRILATERAL-BASED ADAPTIVE MESH

In this paper, the interaction of shock waves with multi-fluids interfaces is investigated by numerical simulations using unstructured quadrilateral adaptive meshes. In order to obtain a detailed structure of the interface, a solution adaptive method for compressible multi-fluid flows developed by Zheng et al. is employed. Firstly, the method is verified by a planar shock and interface interaction problem, which is compared with the front tracking method for the Richtmyer–Meshkov instability problem. Following the verification, the interaction between a circular shock and a sinusoidally perturbed circular interface in cylinder vessel is firstly investigated in our paper. The results show that the solution adaptive method can be employed to study the compressible multi-fluid cases with relatively complex geometry as well as capturing the fine details of interfacial structures of the interaction.

Modified Baer-Nunziato Model for the Simulation of Interfaces Between Compressible Fluids

AbstractIn this paper we proposed a modified Baer-Nunziato model for compressible multi-fluid flows, with main attention on the energy exchange between the two fluids. The proposed model consists of eleven PDEs; however, the use of the particular phase evolving variables may reduce the model to have only six PDEs. The main advantage of the model is that the Abgrall's UPV criterion on mixture velocity and pressure is satisfied without affecting either its hyperbolicity or its conservations of the two individual masses, momentum or total energy. An Lax-Friedrichs scheme is built for a particular case of the proposedmodel. When the two fluids in the fluid mixture are both of the linear Mie-Gruneisen type, the scheme satisfies the Abgrall's UPV criterion on mixture velocity and pressure. Numerical experiments with polytropic, barotropic, stiffened and van der Waals fluids show that the scheme is efficient and able to treat fluids characterized with quite different thermodynamics.

A front-tracking method with projected interface conditions for compressible multi-fluid flows

A front-tracking method for compressible multi-fluid flows is presented, where marker points are used both for tracking fluid interfaces and also for constructing the Riemann problem on the interfaces. The Riemann problem between the two fluid phases (defined in the interface normal direction) is solved using the exact Riemann solver on the marker points. The solutions are projected onto fixed grid points and then extrapolated into the corresponding ghost-fluid regions, to be used as boundary conditions. Each fluid phase is solved separately as in the ghost-fluid method. The proposed procedures, especially the projection of the exact Riemann solutions onto the fluid grids, are designed to be simple and consistent in any spatial dimensions. Several multi-fluid problems, including the breakup of a water cylinder induced by the passage of a shock wave were computed in order to demonstrate the capability of the proposed method.

On the HLLC Riemann solver for interface interaction in compressible multi-fluid flow

In this work, the HLLC Riemann solver, which is much more robust, simpler and faster than iterative Riemann solvers, is extended to obtain interface conditions in sharp-interface methods for compressible multi-fluid flows. For interactions with general equations of state and material interfaces, a new generalized Roe average is proposed. For single-phase interactions, this new Roe average does not introduce artificial states and satisfies the U-property exactly. For interactions at material interfaces, the U-property is satisfied by introducing ghost states for the internal energy. A number of numerical tests suggest that the proposed Riemann solver is suitable for general equations of state and has an accuracy comparable to iterative Riemann solvers, while being significantly more robust and efficient.

Weak solutions for immiscible compressible multifluid flows in porous media

We consider a multifluid flow model describing evolution of pressures and relative saturations of non-miscible and compressible phases in porous media. This model is obtained by a macroscopic representation of the flow. It takes into account capillary effects and velocity fields are described by Darcy laws. Global weak solutions for such a model is introduced. To cite this article: C. Galusinski, M. Saad, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

An object-oriented and quadrilateral-mesh based solution adaptive algorithm for compressible multi-fluid flows

In this paper, an object-oriented and quadrilateral-mesh based solution adaptive algorithm for the simulation of compressible multi-fluid flows is presented. The HLLC scheme (Harten, Lax and van Leer approximate Riemann solver with the Contact wave restored) is extended to adaptively solve the compressible multi-fluid flows under complex geometry on unstructured mesh. It is also extended to the second-order of accuracy by using MUSCL extrapolation. The node, edge and cell are arranged in such an object-oriented manner that each of them inherits from a basic object. A home-made double link list is designed to manage these objects so that the inserting of new objects and removing of the existing objects (nodes, edges and cells) are independent of the number of objects and only of the complexity of O ( 1 ). In addition, the cells with different levels are further stored in different lists. This avoids the recursive calculation of solution of mother (non-leaf) cells. Thus, high efficiency is obtained due to these features. Besides, as compared to other cell-edge adaptive methods, the separation of nodes would reduce the memory requirement of redundant nodes, especially in the cases where the level number is large or the space dimension is three. Five two-dimensional examples are used to examine its performance. These examples include vortex evolution problem, interface only problem under structured mesh and unstructured mesh, bubble explosion under the water, bubble-shock interaction, and shock-interface interaction inside the cylindrical vessel. Numerical results indicate that there is no oscillation of pressure and velocity across the interface and it is feasible to apply it to solve compressible multi-fluid flows with large density ratio (1000) and strong shock wave (the pressure ratio is 10,000) interaction with the interface.

A thermodynamically consistent and fully conservative treatment of contact discontinuities for compressible multicomponent flows

A thermodynamically consistent and fully conservative (TCFC) treatment of contact discontinuities (designated as TCFC model in this paper) is proposed for the simulation of compressible multicomponent flows with shock-interface interactions, yielding an accurate capture of contact discontinuities. Starting from the total energy equation of the mixture, a new formulation is developed to define the ratio of specific heats of the mixture, and a governing equation in conservative form for the pressure is subsequently obtained. Another formulation is derived to determine the molecular weight of the mixture from classical thermodynamics. The governing equations in conservative form are further derived to calculate the specific heats ratio and the molecular weight of the mixture with their new formulations and the continuity or mass fraction equations for the individual components. Finally, the governing equations for the ratio of specific heats, the molecular weight and the pressure, combined with the continuity and momentum equations, offer the new hyperbolic conservation laws for the description of multicomponent or multifluid flows with shock-interface interactions. A simple version of the model is also developed without solving the governing equation for the molecular weight of the mixture. A brief analysis of the numerical uncertainties of conventional conservation models and the present TCFC models is presented and the consistency of any additional equation derived within any model with the initial governing equations is investigated. An analysis of the proposed TCFC model and the conventional conservative models is also performed on the pressure evolution for a mixture associated with isolated contact discontinuities. The results show that the present TCFC model produces a uniform pressure field and thus the pressure maintains in equilibrium across the material interfaces. The proposed TCFC model is then implemented into a Godunov-type method based upon a fast, exact Riemann solver. Several multicomponent flow problems with both strong and weak shocks are simulated using this new model and conventional conservation models, and comparisons of the numerical results from the various models and, where possible, exact solutions are performed. It is shown that the proposed TCFC model can handle both strong and weak shocks, and that the numerical solutions are completely oscillation-free through the contact discontinuities. This new approach offers a suitable treatment of contact discontinuities for compressible multicomponent flows and also maintains all the favorable features of fully conservative conservation laws. It belongs to the conservative approaches and is independent of numerical schemes. From the analysis of the numerical uncertainties of several models proposed in the past and the present numerical experiments, it is determined that the conservative or non-conservative form of some additional governing equations is not the ultimate reason that produces oscillating solutions near the material interfaces. The main cause of the oscillating solutions with conventional conservation models is the inappropriate determination of the ratio of specific heats of the mixture. Oscillation-free results are obtained for two test cases with weak shocks even with conventional conservation models using the present Godunov-type method based on a fast exact Riemann solver. This suggests that the problems with weak shocks and strong post-shock contact discontinuities are not suitable for the model validation in multicomponent flows. The capability of handling strong shocks and weak post-shock contact discontinuities is more critical to the demonstration of any model for compressible multicomponent or multifluid flows.

On the computation of multi-material flows using ALE formulation

Computation of compressible multi-fluid flows with a general equation of state using interface tracking and moving grid approach is discussed in this paper. The AUSM+, HLLC, and Godunov methods are presented and implemented in the context of arbitrary Lagrangian–Eulerian formulation for solving the unsteady compressible Euler equations. The developed methods are fully conservative, and used to compute a variety of multi-component flow problems, where the equations of state can be drastically different and stiff. Numerical results indicate that both ALE HLLC and Godunov schemes demonstrate their simplicity and robustness for solving such multi-phase flow problems, and yet ALE AUSM+ scheme exhibits strong oscillations around material interfaces even using a first order monotone scheme and therefore is not suitable for this class of problems.