We present in this work a new reconstruction scheme, so-called MUSCL-THINC-BVD scheme, to solve the five-equation model for interfacial two phase flows. This scheme employs the traditional shock ...capturing MUSCL (Monotone Upstream-centered Schemes for Conservation Law) scheme as well as the interface sharpening THINC (Tangent of Hyperbola for INterface Capturing) scheme as two building-blocks of spatial reconstruction on the BVD (boundary variation diminishing) principle that minimizes the variations (jumps) of the reconstructed variables at cell boundaries, and thus effectively reduces the dissipation error in numerical solutions. The MUSCL-THINC-BVD scheme is implemented to the volume fraction and other state variables under the same finite volume framework, which realizes the consistency among volume fraction and other physical variables. Numerical results of benchmark tests show that the present method is able to capture the material interface as a well-defined sharp jump in volume fraction, and obtain numerical solutions of superior quality in comparison to other existing methods. The proposed scheme is a simple and effective method of practical significance for simulating compressible interfacial multiphase flows.
•A novel paradigm of spatial reconstruction for compressible multi-phase flows with free interfaces.•Consistency among volume fraction and physical variables across moving material interfaces.•Well resolved moving interface free from numerical dissipation and smearing.•Algorithmic simplicity, computational efficiency and practical significance.•Superior numerical results for wide range benchmark tests to other existing methods.
Maritime structures in heavy seas can experience wave impact events with high loads. The loads can lead to structural failure and even loss of life. Wave breaking in said sea states causes air to be ...entrained in water as aeration cloads, remaining long enough to be transported and to play a role in the impulsive interaction with the structure. A small amount of air in water already forms a highly compressible mixture. Compressibility influences the magnitude of the impact loads.
A new cartesian grid method for compressible multiphase flow is introduced to account for water, air and homogeneous mixtures of air and water. The method is designed to predict the hydrodynamic loads on moving bodies engaging with interfaces between fluids having large density ratios. An equation for conservation of energy is omitted by enforcing pressure-density relations. The interface between fluids is transported using a geometric Volume-of-Fluid method. The interface between fluids and structure is taken care of by a cut-cell method. An additional fraction field for the amount of air in water in combination with a new formulation for the multiphase speed of sound prevent overprediction of compressibility by artificial air entrainment.
New experimental data of 2D wedge impacts with aerated water, made available as open data, are presented to demonstrate the validity of the numerical method. For low aeration levels, the simulation results in terms of the impact loads on the wedge and the frequencies of pressure waves generated upon impact are in good agreement with the experimental data. Increasing the level of aeration reduces the maximum impact load on the wedge. Reflected density waves lead to secondary loads on the wedge. The intensity of the secondary loads, relative to the primary load of impact, increases with the aeration level while the density wave frequency decreases.
•The article features a new compressible, pressure-based multiphase model for the interaction of large-density-ratio flows with a structure.•Representation of impact reduction through cushioning and of propagating density waves in the dispersed air-water mixture.•Verification by means of several benchmarks for compressible media from literature.•Validation by means of new 2D experimental data of wedge entries in aerated water with a free surface, with impact velocities up to Mach 0.5.•Effect of aeration for wedge entries: lower impact pressure for increasing levels of aeration, and density waves leading to pressure oscillations.
In this study, the thermodynamic behavior and supercavitating flow around projectiles of high-subsonic to supersonic speeds were numerically analyzed using a fully compressible multiphase flow with ...phase change. The mathematical model is formulated based on the typical conservation laws of mixtures, including mass, momentum, and energy balance, and by maximizing the thermodynamic properties of water and vapor to enable the analysis of the characteristics of high-speed flows. The model is solved on a body-fitted grid using a finite-volume Riemann solver combined with a monotonic upstream-centered scheme for conservation laws. The numerical method was validated by comparing the supercavitating flow around an underwater projectile with the experimental results available in the literature. The temperature, shock, velocity, and supercavity around the high-speed projectile and the effects of the moving velocities on the physical aspects were investigated. It was observed that the temperature around the stagnation point increased significantly with the increase in the stream velocity, and such high temperatures may soften the steel encasing of the projectile in a manner that the excessive pressure in the area could damage the projectile. The physical context and thermodynamics around the transonic projectile were further observed through numerical investigation.
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•Fully compressible multiphase model for high-speed supercavitation.•Effective non-reflecting boundary condition.•Computations of high-subsonic to supersonic speed projectiles.•Detailed analyses of the thermodynamic behavior of supercavitating flows.
•Two novel strongly hyperbolic reformulations of a hyperbolic model for surface tension.•Godunov-Powell-type formulation of hyperbolic PDE with curl-type involutions.•Generalized Lagrangian ...multiplier (GLM) approach for hyperbolic PDE with curl-type involutions.•Long-time simulations of stationary and oscillating bubbles and shock-bubble interaction.•High order ADER discontinuous Galerkin schemes with a posteriori subcell finite volume limiter.
In this work, we introduce two novel reformulations of the weakly hyperbolic model for two-phase flow with surface tension, recently forwarded by Schmidmayer et al. In the model, the tracking of phase boundaries is achieved by using a new vector field, rather than a scalar tracer, so that the surface-force stress tensor can be expressed directly as an algebraic function of the state variables, without requiring the computation of gradients of the scalar tracer. An interesting and important feature of the model is that this interface field obeys a curl involution constraint, that is, the vector field is required to be curl-free at all times.
The proposed modifications are intended to restore the strong hyperbolicity of the model, and are closely related to divergence-preserving numerical approaches developed in the field of numerical magnetohydrodynamics (MHD). The first strategy is based on the theory of Symmetric Hyperbolic and Thermodynamically Compatible (SHTC) systems forwarded by Godunov in the 60s and 70s and yields a modified system of governing equations which includes some symmetrisation terms, in analogy to the approach adopted later by Powell et al. in the 90s for the ideal MHD equations. The second technique is an extension of the hyperbolic Generalized Lagrangian Multiplier (GLM) divergence cleaning approach, forwarded by Munz et al. in applications to the Maxwell and MHD equations.
We solve the resulting nonconservative hyperbolic partial differential equation (PDE) systems with high order ADER-WENO Finite Volume and ADER Discontinuous Galerkin (DG) methods with a posteriori Finite Volume subcell limiting and carry out a set of numerical tests concerning flows dominated by surface tension as well as shock-driven flows. We also provide a new exact solution to the equations, show convergence of the schemes for orders of accuracy up to ten in space and time, and investigate the role of hyperbolicity and of curl constraints in the long-term stability of the computations.
•A pressure-based methodology for low-Mach flows with mass transfer is presented.•The methodology solves a novel four-equation two-phase diffuse interface model.•Effects of viscosity, surface ...tension, thermal conductivity and gravity are added.•High numerical efficiency due to the use of high-performance solvers is evidenced.•The potential to simulate complex flows is demonstrated with a nucleate boiling case.
This study presents a novel pressure-based methodology for the efficient numerical solution of a four-equation two-phase diffuse interface model. The proposed methodology has the potential to simulate low-Mach flows with mass transfer. In contrast to the classical conservative four-equation model formulation, the adopted set of equations features volume fraction, temperature, velocity and pressure as the primary variables. The model includes the effects of viscosity, surface tension, thermal conductivity and gravity, and has the ability to incorporate complex equations of state. Additionally, a Gibbs free energy relaxation procedure is used to model mass transfer. A key characteristic of the proposed methodology is the use of high performance and scalable solvers for the solution of the Helmholtz equation for the pressure, which drastically reduces the computational cost compared to analogous density-based approaches. We demonstrate the capabilities of the methodology to simulate flows with large density and viscosity ratios through extended verification against a range of different test cases. Finally, the potential of the methodology to tackle challenging phase change flows is demonstrated with the simulation of three-dimensional nucleate boiling.
We present a coupled Eulerian–Lagrangian method to simulate cloud cavitation in a compressible liquid. The method is designed to capture the strong, volumetric oscillations of each bubble and the ...bubble-scattered acoustics. The dynamics of the bubbly mixture is formulated using volume-averaged equations of motion. The continuous phase is discretized on an Eulerian grid and integrated using a high-order, finite-volume weighted essentially non-oscillatory (WENO) scheme, while the gas phase is modeled as spherical, Lagrangian point-bubbles at the sub-grid scale, each of whose radial evolution is tracked by solving the Keller–Miksis equation. The volume of bubbles is mapped onto the Eulerian grid as the void fraction by using a regularization (smearing) kernel. In the most general case, where the bubble distribution is arbitrary, three-dimensional Cartesian grids are used for spatial discretization. In order to reduce the computational cost for problems possessing translational or rotational homogeneities, we spatially average the governing equations along the direction of symmetry and discretize the continuous phase on two-dimensional or axi-symmetric grids, respectively. We specify a regularization kernel that maps the three-dimensional distribution of bubbles onto the field of an averaged two-dimensional or axi-symmetric void fraction. A closure is developed to model the pressure fluctuations at the sub-grid scale as synthetic noise. For the examples considered here, modeling the sub-grid pressure fluctuations as white noise agrees a priori with computed distributions from three-dimensional simulations, and suffices, a posteriori, to accurately reproduce the statistics of the bubble dynamics. The numerical method and its verification are described by considering test cases of the dynamics of a single bubble and cloud cavitation induced by ultrasound fields.
•Mixture-averaged equations of motion are discretized on an Eulerian grid and the individual bubbles are tracked as Lagrangian particles at the sub-grid scale.•The method is capable of resolving fine structures of the strong, bubble-scattered pressure waves.•Model reduction was achieved for bubble clouds possessing translational and rotational homogeneities.•The method is used to simulate a challenging test case of cloud cavitation excited by a strong ultrasound wave.
•An accurate shock- and interface-capturing method for simulations of compressible multiphase flows.•A five-equation two-phase model was transformed into a multi-dimensional general curvilinear ...coordinate system.•High-resolution Godunov-type numerical scheme is constructed using a new eigensystem.•Model is validated by compressible multiphase flow problems under the presence of strong shock waves.•Simulations of 3D complex near-field underwater explosions are presented.
In this study, an accurate shock- and interface-capturing method using curvilinear body-fitted structured grids is introduced to simulate compressible multiphase flows with shockwaves. A five-equation model—proficient in capturing unsteady shocks in compressible multiphase flows without nonphysical spurious oscillations—was enhanced by extending it to a multidimensional general curvilinear coordinate system. A new eigensystem was constructed for a monotonic upstream-centered scheme for conservation laws/Godunov-type finite volume scheme that can capture strong shocks in compressible multiphase flows. The developed method was validated using a typical test involving a free-field underwater explosion that produces a strong shockwave in addition to bubble interfaces. Numerical results were compared and found to agree well with the empirical equation and previously published results. Further, the interaction of a shockwave with a wedge was computed to evaluate the shockwave propagation characteristics and appropriate treatment of solid wall boundary conditions. The numerical results show good agreement with the experimental data. Finally, the impact and propagation characteristics of shockwaves in two more complex cases, involving one and two explosions near a rigid cylinder, were numerically analyzed, and the results indicate high impacts of primary blast waves on the structure.
An accurate treatment of material interfaces in compressible multiphase flows poses important challenges for high-resolution numerical methods. Although high-order interface-capturing schemes have ...been used to accurately simulate gas/liquid interfaces with the Euler equations, these methods can result in temperature spikes at material discontinuities. While this phenomenon is not problematic for Euler simulations, it gives rise to numerical errors when heat conduction is included. In this work, we identify the source of these errors and propose a methodology to prevent their occurrence for various models used to represent gas/liquid interfaces in compressible flows based on a "single-fluid" formulation, in which interfaces are represented by discontinuities in the material properties. Our focus lies in materials (gases and liquids primarily, but also solids) that can be described by a stiffened equation of state, though our approach is generalizable to other equations. We show that numerical approaches that prevent pressure oscillations at interfaces may generate temperature errors, which affect the energy (and pressure) through the heat conduction term. We demonstrate that the material properties entering the equation of state must be computed according to suitable transport equations in conservative or non-conservative forms; the pressure and temperature must be calculated based on the appropriate properties. To verify the analysis and compute problems with gas/liquid interfaces of relevance, we develop a three-dimensional, high-order accurate, solution-adaptive finite difference framework. In particular, we show that temperatures and pressures may be significantly overestimated in calculations of shock-induced bubble collapse in water if temperature errors are not prevented.
Summary
The development of numerical approaches to perform direct numerical simulations of compressible multiphase flows has been an active field of research for several years. Proper treatment of ...fluid interfaces is crucial as important physics occur in this infinitesimally small region. Furthermore, the compressibility of the fluid requires proper treatment of discontinuities. Artificial diffusivity is among a number of methods widely used for compressible flows. This study develops a general form of consistent artificial diffusion fluxes and extends the localized artificial diffusivity method for high‐order central schemes to solve multiphase flows with an interface‐capturing method. These fluxes ensure an oscillation‐free interface for pressure, velocity, and temperature without employing a sharpening technique. Moreover, the high‐order representation of all scales in the flow helps capture the wide range of instabilities inherent in these flows. The goal is to develop an approach capable of performing high‐fidelity simulations supported by physics‐driven validation. This is achieved by solving the five‐equation model with the stiffened‐gas equation of state using the proposed method for multicomponent and multiphase flows on a variety of 1D and 2D problems.
A localized artificial diffusivity method is developed to simulate compressible multifluid systems using central schemes. This is achieved by deriving consistent diffusion fluxes for conservation equations when using the stiffened‐gas equation of state. High‐accuracy simulations comparable with WENO are performed and demonstrate the method's ability to capture a wide range of scales.
•A novel numerical strategy is proposed for the modeling of primary atomization in an unstructured finite volume platform.•An experimental device of a cryogenic flame with subcritical oxygen ...injection is reproduced by Large-Eddy Simulation.•The behavior of physical phenomena from primary atomization of the liquid jet to the combustion of the spray are highlighted.•Both the dynamics of the two-phase flow and the features of the flame are in good agreement with experimental measurements.
Atomization of liquid jets is a key feature of many propulsion systems, such as jet engines, internal combustion engines or liquid-propellant rocket engines (LRE). As it controls the characteristics of the spray, atomization has a great influence on the complex interaction between phenomena such as evaporation, turbulence, acoustics and combustion. In this context, Computational Fluid Dynamics is a promising way to bring better understanding of dynamic phenomena involving atomization, such as e.g. high-frequency combustion instabilities in LRE. However the unsteady simulation of primary atomization in reactive compressible two-phase flows is very challenging, due to the variety of the spatial and temporal scales, as well as to the high density, velocity and temperature gradients which require robust and efficient numerical methods. To address this issue, a numerical strategy is proposed in this paper, which is able to describe the dynamics of the whole chain of mechanisms from the liquid injection to its atomization and combustion. Primary atomization is modeled by a coupling between a homogeneous diffuse interface model and a kinetic-based Eulerian model for the spray. This strategy is successfully applied to the unsteady simulation of an operating point of the Onera’s Mascotte test bench, representative of one coaxial injector of LRE operating under subcritical conditions. The dynamics of the liquid core is retrieved and the flame shape as well as Sauter mean diameters are in good agreement with experimental results. These results demonstrate the ability of the strategy to deal with the harsh conditions of cryogenic combustion, and provide a promising framework for future studies of combustion instabilities in LRE.
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