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The stability of most surface-tension-driven interfacial flow simulations is governed by the capillary time-step constraint. This concerns particularly small-scale flows and, more generally, highly-resolved liquid-gas simulations with moderate inertia. To date, the majority of interfacial-flow simulations are performed using an explicit surface-tension treatment, which restrains the performance of such simulations. Recently, an implicit treatment of surface tension able to breach the capillary time-step constraint using the volume-of-fluid (VOF) method was proposed, based on a fully-coupled pressure-based finite-volume algorithm. To this end, the interface-advection equation is incorporated implicitly into the linear flow solver, resulting in a tight coupling between all implicit solution variables (colour function, pressure, velocity). However, this algorithm is limited to uniform density and viscosity fields. Here, we present a fully-coupled algorithm for interfacial flows with implicit surface tension applicable to interfacial flows with large density and viscosity ratios. This is achieved by solving the continuity and momentum equations in conservative form, whereby the density is treated implicitly with respect to the colour function, and the advection term of the interface-advection equation is discretised using the THINC/QQ algebraic VOF scheme, yielding a consistent discretisation of the advective terms. This new algorithm is tested by considering representative surface-tension-dominated interfacial flows, including the Laplace equilibrium of a stationary droplet and the three-dimensional Rayleigh-Plateau instability of a liquid filament. The presented results demonstrate that interfacial flows with large density and viscosity ratios can be simulated and energy conservation is ensured, even with a time step larger than the capillary time-step constraint.

A significant computational expense and source of numerical errors in front tracking is the remeshing of the triangulated front, required due to distortion and compaction of the front following the Lagrangian advection of its vertices. Additionally, in classic front tracking, the remeshing of the front mesh is required not only due to the deformation of the front shape, but also because the vertices of the front are translated in the direction tangential to the front, induced by the front advection. We present the normal-only advection (NOA) front-tracking method with the aim of preventing the tangential motion of the front vertices and the associated vertex clustering, in order to reduce the number of remeshing operations required to retain a high-quality triangulated interface. To this end, we reformulate the velocity used to advect the front at each discrete front-vertex position. The proposed method is validated and tested against the classic front-tracking method, comparing volume conservation, shape preservation, computational costs, and the overall need for front remeshing, as well as experimental results for canonical interfacial flows. The presented results demonstrate that the NOA front-tracking method leads to a typical reduction of remeshing operations by 80 % or more compared to the classic front-tracking method for well-resolved cases, and results in a smoother front mesh, which is essential for an accurate representation of the geometrical properties of the front. The volume conservation error is reduced by approximately one order of magnitude with the proposed method compared to the classic front-tracking method, at a similar computational cost.

The capillary time-step constraint is the dominant limitation on the applicable time-step in many simulations of interfacial flows with surface tension and, consequently, governs the execution time of these simulations. We propose a fully-coupled pressure-based algorithm based on an algebraic Volume-of-Fluid (VOF) method in conjunction with an implicit linearised surface tension treatment that can breach the capillary time-step constraint. The advection of the interface is solved together with the momentum and continuity equations of the interfacial flow in a single system of linearised equations, providing an implicit coupling between pressure, velocity and the VOF colour function used to distinguish the interacting fluids. Surface tension is treated with an implicit formulation of the Continuum Surface Force (CSF) model, whereby both the interface curvature and the gradient of the colour function are treated implicitly with respect to the colour function. The presented results demonstrate that a time-step larger than the capillary time-step can be applied with this new numerical framework, as long as other relevant time-step restrictions are satisfied, including a time-step restriction associated with surface tension, density as well as viscosity.

We present an explicit finite difference time domain method to solve the lossless Westervelt equation for a moving wave emitting boundary in one dimension and in spherical symmetry. The approach is based on a coordinate transformation between a moving physical domain and a fixed computational domain. This allows to simulate the combined effects of wave profile distortion due to the constitutive nonlinearity of the medium and the nonlinear Doppler modulation of a pressure wave due to the acceleration of the wave emitting boundary. A predictor-corrector method is employed to enhance the numerical stability of the method in the presence of shocks and grid motion. It is demonstrated that the method can accurately predict the Doppler shift of nonlinear wave distortion and the amplitude modulation caused by an oscillating motion of the wave emitting boundary. The novelty of the presented methodology lies in its capability to reflect the Doppler shift of the rate of nonlinear wave profile distortion and shock attenuation for finite amplitude acoustic waves emitted from an accelerating boundary.

Volume conservation and shape preservation are two well-known issues related to the advection and remeshing in front tracking. To address these issues, this paper proposes a divergence-preserving velocity interpolation method and a parabolic fit vertex positioning method for remeshing operations for three-dimensional front tracking. Errors in preserving the divergence of the velocity field when interpolating the velocity from the fluid mesh to the vertices of the triangles of the front are a primary reason for volume conservation errors when advecting the front. The proposed interpolation method preserves the discrete divergence of the fluid velocity by construction and is compared in this work with other known interpolation methods in divergence-free and non-divergence-free test cases, with respect to volume conservation and shape preservation of the front. The presented interpolation method conserves the volume and shape up to an order of magnitude better than the conventionally used interpolation methods and is within the range of higher order interpolation methods at lower computational cost. Additionally, the parabolic fit vertex positioning method for remeshing operations locally approximates the front with a smooth polynomial surface, improving volume conservation and shape preservation by an order of magnitude compared to conventional remeshing algorithms.

The majority of available numerical algorithms for interfacial two-phase flows either treat both fluid phases as incompressible (constant density) or treat both phases as compressible (variable density). This presents a limitation for the prediction of many two-phase flows, such as subsonic fuel injection, as treating both phases as compressible is computationally expensive due to the very stiff pressure-density-temperature coupling of liquids. A framework with the capability of treating one phase compressible and the other phase incompressible, therefore, has a significant potential to improve the computational performance and still capture all important physical mechanisms. We propose a numerical algorithm that can simulate interfacial flows in all Mach number regimes, ranging from $M=0$ to $M > 1$, including interfacial flows in which compressible and incompressible fluids interact, within the same pressure-based framework and conservative finite-volume discretisation. For interfacial flows with only incompressible fluids or with only compressible fluids, the proposed pressure-based algorithm and finite-volume discretisation reduce to numerical frameworks that have already been presented in the literature. Representative test cases are used to validate the proposed algorithm, including mixed compressible-incompressible interfacial flows with acoustic waves, shock waves and rarefaction fans.

The particle-source-in-cell Euler-Lagrange (PSIC-EL) method is widely used to simulate flows laden with particles. Its accuracy, however, is known to deteriorate as the ratio between the particle diameter~($\smash{d_\text{p}}$) and the mesh spacing~($h$) increases, due to the impact of the momentum that is fed back to the flow by the Lagrangian particles. Although the community typically recommends particle diameters to be at least an order of magnitude smaller than the mesh spacing, the errors corresponding to a given $\smash{d_\text{p} / h}$ ratio and/or flow regime have not been systematically studied. In~this paper, we provide an expression to estimate the magnitude of the flow velocity disturbance resulting from the transport of a particle in the PSIC-EL framework, based on the $\smash{d_\text{p} / h}$ ratio and the particle Reynolds number, $\smash{\text{Re}_\text{p}}$. This, in turn, directly relates to the error in the estimation of the undisturbed velocity, and therefore to the error in the prediction of the particle motion. We show that the upper bound of the relative error in the estimation of the undisturbed velocity, for all particle Reynolds numbers, is approximated by $\smash{(6/5)\,d_\text{p} / h}$. Moreover, for all cases where $\smash{d_\text{p} / h \lesssim 1/2}$, the expression we provide accurately estimates the value of the errors across a range of particle Reynolds numbers that are relevant to most gas-solid flow applications ($\smash{\text{Re}_\text{p} < 500}$).

The Gilmore model is combined with the Noble-Abel-stiffened-gas (NASG) equation of state to yield a simple model to predict the expansion and collapse of spherical bubbles based on real gas thermodynamics. The NASG equation of state resolves the temperature inaccuracy associated with the commonly employed Tait equation of state for liquids and, thus, can provide a consistent description of compressible and thermal effects of the bubble content and the surrounding liquid during cavitation. After a detailed derivation of the proposed Gilmore-NASG model, the differences between the classical Gilmore-Tait model and the proposed model are highlighted with results of single-bubble cavitation related to bubble collapse and driven by an acoustic excitation in frequency and amplitude regimes relevant to sonoluminescence, high-intensity focused ultrasound and shock wave lithotripsy. Especially for rapidly and violently collapsing bubbles, substantial differences in the bubble behaviour can be observed between the proposed Gilmore-NASG model and the classical Gilmore-Tait model. The ability of the Gilmore-NASG model to simultaneously predict reliable pressure and temperature values in gas, vapour and liquid, makes the proposed model particularly attractive for sonochemistry and biomedical applications.

A conservative finite-volume framework, based on a collocated variable arrangement, for the simulation of flows at all speeds, applicable to incompressible, ideal-gas and real-gas fluids is proposed in conjunction with a fully-coupled pressure-based algorithm. The applied conservative discretisation and implementation of the governing conservation laws as well as the definition of the fluxes using a momentum-weighted interpolation are identical for incompressible and compressible fluids, and are suitable for complex geometries represented by unstructured meshes. Incompressible fluids are described by predefined constant fluid properties, while the properties of compressible fluids are described by the Noble-Abel-stiffened-gas model, with the definitions of density and specific static enthalpy of both incompressible and compressible fluids combined in a unified thermodynamic closure model. The discretised governing conservation laws are solved in a single linear system of equations for pressure, velocity and temperature. Together, the conservative finite-volume discretisation, the unified thermodynamic closure model and the pressure-based algorithm yield a conceptually simple, but versatile, numerical framework. The proposed numerical framework is validated thoroughly using a broad variety of test-cases, with Mach numbers ranging from 0 to 239, including viscous flows of incompressible fluids as well as the propagation of acoustic waves and transiently evolving supersonic flows with shock waves in ideal-gas and real-gas fluids. These results demonstrate the accuracy, robustness and the convergence, as well as the conservation of mass and energy, of the numerical framework for flows of incompressible and compressible fluids at all speeds, on structured and unstructured meshes.

The interaction of a shock wave with a bubble features in many engineering and emerging technological applications, and has been used widely to test new numerical methods for compressible interfacial flows. Recently, density-based algorithms with pressure-correction methods as well as fully-coupled pressure-based algorithms have been established as promising alternatives to classical density-based algorithms based on Riemann solvers. The current paper investigates the predictive accuracy of fully-coupled pressure-based algorithms without Riemann solvers in modelling the interaction of shock waves with one-dimensional and two-dimensional bubbles in gas-gas and liquid-gas flows. For a gas bubble suspended in another gas, the mesh resolution and the applied advection schemes are found to only have a minor influence on the bubble shape and position, as well as the behaviour of the dominant shock waves and rarefaction fans. For a gas bubble suspended in a liquid, however, the mesh resolution has a critical influence on the shape, the position and the post-shock evolution of the bubble, as well as the pressure and temperature distribution.

The impact of different linearisation and iterative solution strategies for fully-coupled pressure-based algorithms for compressible flows at all speeds is studied, with the aim of elucidating their impact on the performance of the numerical algorithm. A fixed-coefficient linearisation and a Newton linearisation of the transient and advection terms of the governing nonlinear equations are compared, focusing on test-cases that feature acoustic, shock and expansion waves. The linearisation and iterative solution strategy applied to discretise and solve the nonlinear governing equations is found to have a significant influence on the performance and stability of the numerical algorithm. The Newton linearisation of the transient terms of the momentum and energy equations is shown to yield a significantly improved convergence of the iterative solution algorithm compared to a fixed-coefficient linearisation, while the linearisation of the advection terms leads to substantial differences in performance and stability at large Mach numbers and large Courant numbers. It is further shown that the consistent Newton linearisation of all transient and advection terms of the governing equations allows, in the context of coupled pressure-based algorithms, to eliminate all forms of underrelaxation and provides a clear performance benefit for flows in all Mach number regimes.