Compressible Two-fluid Flow Research Articles

Analysis on physical-constraint-preserving high-order discontinuous Galerkin method for solving Kapila's five-equation model

This work focuses on the analysis and development of a high-order physical-constraint-preserving discontinuous Galerkin (DG) method for compressible two-fluid flows by solving Kapila's five-equation model with the ideal gas equation of state. The proposed method is proved to possess the following two attractive advantages: first, the high-order DG discretization intrinsically satisfies the mechanical equilibrium criterion around an isolated material interface, and the numerical oscillations due to the use of high-order approximations are well suppressed by using a cell-based characteristic-wise weighted essentially non-oscillatory (WENO) limiter without destroying the mechanical equilibrium criterion; second, it maintains the physical-constraint-preserving property, including the bound-preserving property of volume fractions and positivity-preserving property of partial densities and internal energy. A thorough analysis on the sufficient condition is given to achieve the physical-constraint-preserving property. Several benchmark test cases are examined to validate the accuracy and robustness of the proposed method. Further discussion on the application of the physical-constraint-preserving DG method for solving Kapila's five-equation model with the stiffened gas equation of state is also presented.

An Integrated Modeling Approach for Vertical Gas Migration Along Leaking Wells Using a Compressible Two-Fluid Flow Model

Gas migration behind casings can occur in wells where the annular cement barrier fails to provide adequate zonal isolation. A direct consequence of gas migration is annular pressure build-up at wellhead, referred to as sustained casing pressure (SCP). Current mathematical models for analyzing SCP normally assume gas migration along the cemented interval to be single-phase steady-state Darcy flow in the absence of gravity and use a drift-flux model for two-phase transport through the mud column above the cement. By design, such models do not account for the possible simultaneous flow of gas and liquid along the annulus cement or the impact of liquid saturation within the cemented intervals on the surface pressure build-up. We introduce a general compressible two-fluid model which is solved over the entire well using a newly developed numerical scheme. The model is first validated against field observations and used for a parametric study. Next, detailed studies are performed, and the results demonstrate that the surface pressure build-up depends on the location of cement intervals with low permeability, and the significance of two-phase co-current or counter-current flow of liquid and gas occurs along cement barriers that have an initial liquid saturation. As the magnitude of the frictional pressure gradient associated with counter-current of liquid and gas can be comparable to the relevant hydrostatic pressure gradient, two-phase flow effects can significantly impact the interpretation of the wellhead pressure build-up.

Local well-posedness of solutions to the boundary layer equations for compressible two-fluid flow

<p style='text-indent:20px;'>In this paper, we consider the two-dimensional (2D) two-fluid boundary layer system, which is a hyperbolic-degenerate parabolic-elliptic coupling system derived from the compressible isentropic two-fluid flow equations with nonslip boundary condition for the velocity. The local existence and uniqueness is established in weighted Sobolev spaces under the monotonicity assumption on tangential velocity along normal direction based on a nonlinear energy method by employing a nonlinear cancelation technic introduced in [R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang, J. Amer. Math. Soc., 28 (2015), 745-784; N. Masmoudi and T.K. Wong, Comm. Pure Appl. Math., 68(2015), 1683-1741] and developed in [C.-J. Liu, F. Xie and T. Yang, Comm. Pure Appl. Math., 72(2019), 63-121].</p>

A monotonicity-preserving higher-order accurate finite-volume method for Kapila’s two-fluid flow model

In preparation of the study of liquefied natural gas (LNG) sloshing in ships and vehicles, we model and numerically analyze compressible two-fluid flow. We consider a five-equation two-fluid flow model, assuming velocity and pressure continuity across two-fluid interfaces, with a separate equation to track the interfaces. The system of partial differential equations is hyperbolic and quasi-conservative. It is discretized in space with a tailor-made third-order accurate finite-volume method, employing an HLLC approximate Riemann solver. The third-order accuracy is obtained through spatial reconstruction with a limiter function, for which a novel formulation is presented. The non-homogeneous term is handled in a way consistent with the HLLC treatment of the convection operator. We study the one-dimensional case of a liquid column impacting onto a gas pocket entrapped at a solid wall. It mimics the impact of a breaking wave in an LNG containment system, where a gas pocket is entrapped at the tank wall below the wave crest. Furthermore, the impact of a shock wave on a gas bubble containing the heavy gas R22, immersed in air, is simulated in two dimensions and compared with experimental results. The numerical scheme is shown to be higher-order accurate in space and capable of capturing the important characteristics of compressible two-fluid flow.

High order finite-volume WENO scheme for five-equation model of compressible two-fluid flow

High order finite-volume weighted essentially non-oscillatory (WENO) scheme is applied for solving interface-capturing five-equation model of compressible two-fluid flows in one and two-space dimensions. The model is non-conservative and the governing equations consist of three equations, namely a continuity equation, a momentum equation and an energy balance equation for the fluid mixture and the remaining two are mass and energy equations for one of the two fluids. In the last equation, the non-conservative differential source term appears which is responsible for the energy exchange between fluids. The energy exchange is only due to mechanical work. The presence of non-conservative differential source terms in the two-fluid flow model introduce difficulties in developing high order accurate numerical schemes. The proposed numerical scheme is capable to preserve non-oscillatory property near strong discontinuities and gives high order accuracy in smooth regions. Different one and two-dimensional test problems are considered to analyze efficiency and accuracy of the proposed numerical algorithms. For validation, the solutions of proposed numerical scheme are compared with the results of already available high order kinetic flux-vector splitting scheme and discontinuous Galerkin scheme.

A high-resolution method for compressible two-fluid flows and simulation of three-dimensional shock-bubble interactions

SUMMARY A high-resolution method is developed to capture the material interfaces of compressible two-fluid flows in multiple dimensions. A fluid mixture model system with single velocity and pressure is used, and viscous effect can also be taken into account. A consistent thermodynamic law based on the assumption of pressure equilibrium is employed to describe the thermodynamic behaviors of the pure fluids and mixture of two components. The splitting and unsplit Eulerian formulations of piecewise parabolic method are extended to numerically integrate the hyperbolic part of the model system, whereas the system of diffusion equations is solved using an explicit, central difference scheme. The block-structured adaptive mesh refinement (AMR) capability is built in the hydrodynamic code to locally improve grid resolution. The resulting method is verified to be at least second-order accurate in space. Numerical results show that the discontinuities, particularly contact discontinuities, can be resolved sharply. The use of AMR allows flow features at disparate scales to be resolved sufficiently. In addition, three-dimensional shock–bubble interactions are simulated to investigate effects of Mach number on bubble evolution. The flow structures including those peculiar to three-dimensional bubble are resolved correctly, and some physical phenomena with increasing Mach number are reported. Copyright © 2012 John Wiley & Sons, Ltd.

The Space-Time CE/SE Method for Solving Reduced Two-Fluid Flow Model

AbstractThe space-time conservation element and solution element (CE/SE) method is proposed for solving a conservative interface-capturing reduced model of compressible two-fluid flows. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term for accounting the energy exchange. The one and two-dimensional flow models are numerically investigated in this manuscript. The CE/SE method is capable to accurately capture the sharp propagating wavefronts of the fluids without excessive numerical diffusion or spurious oscillations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building block of the suggested method. The method differs from the previous techniques because of global and local flux conservation in a space-time domain without resorting to interpolation or extrapolation. In order to reveal the efficiency and performance of the approach, several numerical test cases are presented. For validation, the results of the current method are compared with other finite volume schemes.

A new formulation of Kapila’s five-equation model for compressible two-fluid flow, and its numerical treatment

A new formulation of Kapila’s five-equation model for inviscid, non-heat-conducting, compressible two-fluid flow is derived, together with an appropriate numerical method. The new formulation uses flow equations based on conservation laws and exchange laws only. The two fluids exchange momentum and energy, for which exchange terms are derived from physical laws. All equations are written as a single system of equations in integral form. No equation is used to describe the topology of the two-fluid flow. Relations for the Riemann invariants of the governing equations are derived, and used in the construction of an Osher-type approximate Riemann solver. A consistent finite-volume discretization of the exchange terms is proposed. The exchange terms have distinct contributions in the cell interior and at the cell faces. For the exchange-term evaluation at the cell faces, the same Riemann solver as used for the flux evaluation is exploited. Numerical results are presented for two-fluid shock-tube and shock-bubble-interaction problems, the former also for a two-fluid mixture case. All results show good resemblance with reference results.

Numerical simulation of liquid sloshing in a partially filled container with inclusion of compressibility effects

A numerical scheme of study is developed to model compressible two-fluid flows simulating liquid sloshing in a partially filled tank. For a two-fluid system separated by an interface as in the case of sloshing, not only a Mach-uniform scheme is required but also an effective way to eliminate unphysical numerical oscillations near the interface. By introducing a preconditioner, the governing equations expressed in terms of primitive variables are solved for both fluids (i.e., water, air, gas, etc.) in a unified manner. In order to keep the interface sharp and to eliminate unphysical numerical oscillations in unsteady fluid flows, the nonconservative implicit split coefficient matrix method is modified to construct a flux-difference splitting scheme in the dual-time formulation. The proposed numerical model is evaluated by comparisons between numerical results and measured data for sloshing in an 80% filled rectangular tank excited at resonance frequency. Through similar comparisons, the investigation is further extended by examining sloshing flows excited by forced sway motions in two different rectangular tanks with 20% and 83% filling ratios. These examples demonstrate that the proposed method is suitable to capture induced free surface waves and to evaluate sloshing pressure loads acting on the tank walls and ceiling.

A high order kinetic flux-vector splitting method for the reduced five-equation model of compressible two-fluid flows

We present a high order kinetic flux-vector splitting (KFVS) scheme for the numerical solution of a conservative interface-capturing five-equation model of compressible two-fluid flows. This model was initially introduced by Wackers and Koren (2004) [21]. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term in order to account for the energy exchange. We numerically investigate both one- and two-dimensional flow models. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the system of equations. In two space dimensions the scheme is derived in a usual dimensionally split manner. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge–Kutta time stepping method. For validation, the results of our scheme are compared with those from the high resolution central scheme of Nessyahu and Tadmor [14]. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential for modeling two-phase flows.

An Interface-Capturing Method for Resolving Compressible Two-Fluid Flows with General Equation of State

An Interface-Capturing Method for Resolving Compressible Two-Fluid Flows with General Equation of State

Numerical simulation of Richtmyer–Meshkov instability driven by imploding shocks

In this paper, the classical piecewise parabolic method (PPM) is generalized to compressible two-fluid flows, and is applied to simulate Richtmyer–Meshkov instability (RMI) induced by imploding shocks. We use the compressible Euler equations together with an advection equation for volume fraction of one fluid component as model system, which is valid for both pure fluid and two-component mixture. The Lagrangian-remapping version of PPM is employed to solve the governing equations with dimensional-splitting technique incorporated for multi-dimensional implementation, and the scheme proves to be non-oscillatory near material interfaces. We simulate RMI driven by imploding shocks, examining cases of single-mode and random-mode perturbations on the interfaces and comparing results of this instability in planar and cylindrical geometries. Effects of perturbation amplitude and shock strength are also studied.

Numerical Simulation of Two-Fluid Two-Phase Flows by HLL Scheme Using an Approximate Jacobian Matrix

ABSTRACT A compressible two-fluid two-phase flow model based on two sets of governing equations is formulated. We solve, using the second-order Harten, Lax, and van Leer (HLL) scheme, five disparate two-phase benchmark problems such as shock propagation in the two-phase medium, the cavitating flow, and the gravity-induced phase separation. In the conventional HLL scheme, the simple sonic speeds evaluated from the two single phases are used in lieu of the fastest speeds in the two phases, since their accurate estimation is difficult. To improve the method, we here propose utilization of the analytic eigenvalues from the temporary 6 × 6 Jacobian matrix, reduced by dropping the interfacial transfer terms. The total sound speed of the two-phase flow evaluated by these eigenvalues agrees very well with the existing experimental data. The second-order HLL scheme using these analytic eigenvalues is proved efficient, robust, and accurate in comparison with other available methods.