Compressible Two-phase Model Research Articles

Numerical investigation on the liquid jet and the dynamics of the near-wall cavitation bubble under an acoustic field

The dynamics of the near-wall cavitation bubble in an acoustic field are the fundamental forms of acoustic cavitation, which has been associated with promising applications in ultrasonic cleaning, chemical engineering, and food processing. However, the potential physical mechanisms for acoustic cavitation-induced surface cleaning have not been fully elucidated. The dynamics of an ultrasonically driven near-wall cavitation bubble are numerically investigated by employing a compressible two-phase model implemented in OpenFOAM. The corresponding validation of the current model containing the acoustic field was performed by comparison with experimental and state-of-the-art theoretical results. Compared to the state without the acoustic field, the acoustic field can enhance the near-wall bubble collapse due to its stretching effect, causing higher jet velocities and shorter collapse intervals. The jet velocity in the acoustic field increases by 80.2%, and the collapse time reduces by 40.9% compared to those without an acoustic field for γ = 1.1. In addition, the effects of the stand-off distances (γ), acoustic pressure wave frequency (f), and initial pressure (p*) on the bubble dynamic behaviors were analyzed in depth. The results indicate that cavitation effects (e.g., pressure loads at the wall center and the maximal bubble temperature) are weakened with the increase in the frequency (f) owing to the shorter oscillation periods. Furthermore, the maximum radius of bubble expansion and the collapse time decrease with increasing f and increase with increasing p*. The bubble maximum radius reduces by 12.6% when f increases by 62.5% and increases by 20.5% when p* increases by 74%.

Shock wave induced by the collapse of a bubble cluster with each bubble distributed randomly

The phenomenon of bubble collapse is ubiquitous in both natural and industrial settings, often manifesting in the form of bubble clusters. Despite its prevalence, there has been limited research on the dynamics of bubble clusters. In order to gain a better understanding of shock wave emission, liquid jet propagation, and the resulting pressure loads during bubble cluster collapse, we have developed a high-fidelity numerical approach aimed at accurately capturing shock waves and phase interfaces. The interface compression technique is introduced into the compressible two-phase model to improve the sharpness of the phase interface. We solve the physical model using the BVD (Boundary Variation Diminishing) principle. Both the 5th order WENO (Weighted Essentially Non-Oscillatory) scheme and the interface capturing function THINC (Tangent of Hyperbola for INterface Capturing) are utilized to reconstruct primary variables at cell boundaries, effectively minimizing numerical dissipation. For further resolution improvement, we implement these numerical methods on the AMR (Adaptive Mesh Refinement) mesh. Our numerical results demonstrate that the distance between the wall and the bubble cluster, as well as the nearest bubble to the wall, significantly influence the wall peak pressure. However, they have a minimal impact on the overall evolution of the bubble cluster and the peak pressure induced in the flow field. Remarkably, the spatial distribution of bubbles and the total number of bubbles play a crucial role in shaping the dynamic behavior of the bubble cluster and the pressure loads exerted both on the solid wall and in the flow field. The crucial findings offer profound insights into a range of applications, including cavitation erosion, underwater explosions, and drug delivery.

Wave-in-deck loads induced by regular wave impact: The role of compressible air entrainment

During the process of wave-in-deck when waves impacting a horizontal platform deck from below, large air cavity entrainment would be generated between the free wave surface and the deck. The air entrainments are compressed and expanded during the process of wave impacts. As the momentum and energy exchange medium in the interactions between waves and the structure, air entrainments conversely influence the forces on the deck, risking structural safety. The key to dealing with the air entrainment is the combination of the mass source wave method and pressure-based compressible technique in capturing the cavity volume variation. In this work, the advanced compressible two-phase model is first developed with our in-house solver to simulate a regular wave impacting a fixed flat plate. The air cavity kinematics/dynamics and their relationship with wave impact features in the process of wave-in-deck impacting are discussed, and the mechanism of the compression/expansion effect is carefully analysed. Two distinct impacting types, direct water impact and entrained-air impact, are identified in this work, and the larger wave impacting pressures in the latter situation with compressible air layer is demonstrated. The good comparison of the numerical results and laboratory data on pressure features reveals that the proposed model is capable of reproducing the nonlinear entrained-air wave impacts and accurate oscillating pressures on a flat deck for applications.

Dynamics of a Laser-Induced Cavitation Bubble near a Cone: An Experimental and Numerical Study

A bubble’s motion is strongly influenced by the boundaries of tip structures, which correspond to the bubble’s size. In the present study, the dynamic behaviors of a cavitation bubble near a conical tip structure are investigated experimentally and numerically. A series of experiments were carried out to analyze the bubble’s shape at different relative cone distances quantitatively. Due to the crucial influence of the phase change on the cavitation bubble’s dynamics over multiple cycles, a compressible two-phase model taking into account the phase change and heat transfer implemented in OpenFOAM was employed in this study. The simulation results regarding the bubble’s radius and shape were validated with corresponding experimental photos, and a good agreement was achieved. The bubble’s primary physical features (e.g., shock waves, liquid jets, high-pressure zones) were well reproduced, which helps us understand the underlying mechanisms. Meanwhile, the latent damage was quantified by the pressure load at the cone apex. The effects of the relative distance γ and cone angle θ on the maximum temperature, pressure peaks, and bubble position are discussed and summarized. The results show that the pressure peaks during the bubble’s collapse increase with the decrease in γ. For a larger γ, the first minimum bubble radius increases while the maximum temperature decreases as θ increases; the pressure peak at the second final collapse is first less than that at the first final collapse and then much greater than that one. For a smaller γ, the pressure peaks at different θ values do not vary very much.

Asymptotic stability of planar rarefaction wave to a multi-dimensional two-phase flow

<p style='text-indent:20px;'>We are concerned with the time-asymptotic stability of planar rarefaction wave to a non-conservative two-phase flow system described by two-dimentional compressible Euler and Navier-Stokes equations through drag force. In this paper, we show the planar rarefaction wave to a non-conservative compressible two-phase model is asymptotically stable under small initial perturbation in <inline-formula><tex-math id="M1">\begin{document}$ H^3 $\end{document}</tex-math></inline-formula>. The main difficulties overcome in this paper come from the non-viscosity of one fluid and the interaction between two fluids caused by drag force. The stability result is proved by the energy method.</p>

A phase field model for compressible immiscible fluids with a new equation of state

In this paper, we propose a new compressible phase field model of two different immiscible fluid components, in which the density of each phase is variable. In order to establish the compressible phase field model, a new P-V-T equation of state is introduced to solve the pressure. The new model is derived by the physics law of conservation, and conforms to the second law of thermodynamics. The model adopts an innovative expression of Helmholtz free energy, taking into account the new state equation of pressure and the varying material properties of each phase. A high-order accurate numerical scheme is introduced to solve the model equations. The convection terms of the governing equations are discretized by the fifth-order WENO scheme, and the residual terms are discretized by the Lax–Friedrichs method. Finally, the reliability and validity of the compressible two-phase model are verified by numerical simulations.

Stability of planar rarefaction wave to a multi-dimensional non-conservative viscous compressible two-phase flow

We are concerned with the time-asymptotic stability of planar rarefaction wave to a non-conservative viscous compressible two-phase flow in two dimensions. In this paper, we show the planar rarefaction wave to a non-conservative viscous compressible two-phase model is asymptotically stable under small initial perturbation in H2. The stability result is proved by the energy method.

Local exact controllability for a viscous compressible two-phase model

This paper is devoted to the boundary controllability of a viscous compressible two-phase system modeling the motion of the mixture of fluid and particles. Based on the observability inequality of the adjoint system of a new linearized system including the equation of the pressure and the fixed point argument, the local exact controllability of this system on a constant target trajectory is obtained. The initial density of the fluid and the particles may contain vacuum and one of the target density allows to be vanish.

Numerical modeling of multiphase compressible flows with the presence of shock waves using an interface-sharpening five-equation model

An accurate shock- and interface-capturing method is introduced for simulations of compressible multiphase flows. First, an associated Godunov-type numerical scheme is established for a five-equation two-phase model obtained from a seven-equation model by assuming a single velocity and a single pressure between two phases. The computational finite-volume Riemann solver using the scheme and computing algorithms is presented. Next, an interface-sharpening technique (IST) is extended for the compressible two-phase model to improve numerical simulations and correct diffusion errors. The modified IST was applied as postprocessing to correct the numerical diffusion error in the solution of the discretization scheme while maintaining a sharp interface with a desired thickness after each time step. A mixture-consistent interface regularization approach of all conservative variables is combined with the IST to obtain consistent thermodynamic laws for the mixture, ensuring the consistency of the variables in the correction process. Several examples of fluid interface simulations including shock-tube, shock-bubble interactions, and underwater explosions were performed to demonstrate the accuracy and capability of the proposed method. Those compressible multiphase flow problems are complicated by the presence of both shock waves and the dynamics of interfaces. Comparisons of the numerical method with theoretical results and experimental data indicate that the present method can simulate interface dynamics with the presence of shock waves and large density differences.

On the thermodynamic behaviors and interactions between bubble pairs: A numerical approach

Thermodynamic behaviors and interactions between bubble pairs are important to better understand the cavitation phenomena. In this study, a compressible two-phase model, accounting for thermal effects to investigate the thermodynamic behaviors and interactions between bubble pairs, is developed in OpenFOAM. The volume of fluid (VOF) method is adopted to capture the interface. Validations are performed by comparing the simulation results of a single bubble and bubble pairs with corresponding experimental data. The dynamical behaviors of bubble pairs and their thermodynamic effect at different relative distances γ are investigated and discussed, which help reveal the bubble cloud dynamics. The quantitative analysis of γ effects on the maximum temperature during bubble collapse is performed with three distinct stages identified. For a single bubble collapsing near the rigid surface, the thermodynamic characteristics at different relative distances are similar to that of the bubble pairs, but the maximum temperature is higher since the single bubble can collapse to a smaller volume.

A numerical study on the influence of liquid properties on gas-focused micro-jets

In this paper we present a numerical study on the influence of liquid properties on gas-focused micro-jets, such as used for sample delivery in serial femtosecond crystallography. The study is based on solving mixture formulation of Newtonian, compressible two-phase model with the finite volume method and algebraic volume of fluid for treatment of the phase-interface. The density, viscosity and surface tension of the focused fluid span around the material properties of pure water in the range of ±30%, thus representing a large range of possible sample delivery fluids. Fixed liquid and helium gas flow rates are used for jet focusing and the length, diameter, velocity and temperature of the jet are assessed as a function of material properties of the liquid. A thicker and slower jet is observed in case of increased density and surface tension of the focused fluid, while change in liquid viscosity has no effect on flow characteristics.

A compressible two-phase model for dispersed particle flows with application from dense to dilute regimes

Multiphase flows are present in many important fields ranging from multiphase explosions to chemical processing. An important subset of multiphase flow applications involves dispersed materials, such as particles, droplets, and bubbles. This work presents an Eulerian–Eulerian model for multiphase flows containing dispersed particles surrounded by a continuous media such as air or water. Following a large body of multiphase literature, the driving force for particle acceleration is modeled as a direct function of both the continuous-phase pressure gradient and the gradient of intergranular stress existing within the particle phase. While the application of these two components of driving force is well accepted in much of the literature, other models exist in which the particle-phase pressure gradient itself drives particle motion. The multiphase model treats all phases as compressible and is derived to ensure adherence to the 2nd Law of Thermodynamics. The governing equations are presented and discussed, and a characteristic analysis shows the model to be hyperbolic, with a degeneracy in the case that the intergranular stress, which is modeled as a configuration pressure, is zero. Finally, results from a two sample problems involving shock-induced particle dispersion are presented. The results agree well with experimental measurements, providing initial confidence in the proposed model.

Solving seven-equation model for compressible two-phase flow using multiple GPUs

In this paper, the application of an HLLC-type approximate Riemann solver in conjunction with the third-order TVD Runge–Kutta method to the seven-equation compressible two-phase model on multiple Graphics Processing Units (GPUs) is presented. Based on the idea proposed by Abgrall et al. that “a multiphase flow, uniform in pressure and velocity at t=0, will remain uniform on the same variables during time evolution”, discretization schemes for the non-conservative terms and for the volume fraction evolution equation are derived in accordance with the HLLC solver used for the conservative terms. To attain high temporal accuracy, the third-order TVD Runge–Kutta method is implemented in conjunction with operator splitting technique, in which the sequence of operators is recorded in order to compute free surface problems robustly. For large scale simulations, the numerical method is implemented using MPI/Pthread-CUDA parallelization paradigm for multiple GPUs. Domain decomposition method is used to distribute data to different GPUs, parallel computation inside a GPU is accomplished using CUDA, and communication between GPUs is performed via MPI or Pthread. Efficient data structure and GPU memory usage are employed to maintain high memory bandwidth of the device, while a special procedure is designed to synchronize thread blocks so as to reduce frequencies of kernel launching. Numerical tests against several one- and two-dimensional compressible two-phase flow problems with high density and high pressure ratios demonstrate that the present method is accurate and robust. The timing tests show that the overall speedup of one NVIDIA Tesla C2075 GPU is 31× compared with one Intel Xeon Westmere 5675 CPU core, and nearly 70% parallel efficiency can be obtained when using 8 GPUs.

Asymptotic behavior of a compressible two-phase model with well–formation interaction

In this work we consider a compressible gas–liquid model with a well–reservoir interaction term that is relevant for coupled wellbore-reservoir flow systems involved in e.g. drilling operations. Main focus is on deriving estimates that are independent of time. Under suitable conditions on the well–reservoir interaction term we obtain such estimates which allow prediction of the long-time behavior of the gas and liquid masses. Moreover, we also obtain a quantification of the convergence rates as a function of time and gain some insight into the role played by the rate characterizing how fast the well–reservoir interaction must die out. The model is investigated in a free boundary setting where the initial mass is a mixture of both phases, i.e., no single-phase zone exists.

Global Weak Solutions to One-Dimensional Non-Conservative Viscous Compressible Two-Phase System

In this paper, we deal with global weak solutions of a non-conservative viscous compressible two-phase model in one space dimension. This work extends in some sense the previous work, [Bresch et al., Arch Rat Mech Anal 196:599–629, 2010], which provides the global existence of weak solutions in the multi-dimensional framework with 1 1. Then we prove that any possible vacuum state has to vanish within finite time after which densities are always away from vacuum. This allows to prove that at least one phase corresponding to the global weak solution is a locally in time and space (in a sense to be defined) strong solution after the vacuum states vanish. Our paper may be understood as a non-straightforward generalization to the two-phase flow system of a previous paper [Li et al., Commun Math Phys 281(2):401–444, 2008], which treated the usual compressible barotropic Navier-Stokes equations for mono-fluid with a degenerate viscosity. Various important mathematical difficulties occur when we want to generalize those results to the two-phase flows system since the corresponding model is non-conservative. Far from vacuum, it involves a strong coupling between a nonlinear algebraic system and a degenerate PDE system under constraint linked to fractions. Moreover, fractional densities may vanish if densities or fractions vanish: A difficulty is to find estimates on the densities from estimates on fractional densities using the algebraic system. Original approximate systems have also to be introduced compared to the works on the degenerate barotropic mono-fluid system. Note that even if our result concerns “only” the one-dimensional case, it points out possible global weak solutions (for such a non-conservative system) candidates to approach for instance shock structures and to define an appropriate a priori family of paths in the phase space (in numerical schemes) at the zero dissipation limit.

GENERIC Treatment of Compressible Two-Phase Flow: Convection Mechanism of Scalar Morphological Variables

A two-phase model is considered which resolves both the thermodynamics of the two phases and of the interface, as well as the morphology in terms of Minkowski functionals, as scalar morphological variables, assuming equal velocities of the phases. In order to discuss whether, in compressible flow, the morphological variables should transform as scalars, as scalar densities or as a mixture of both, the two-phase model is cast into the GENERIC framework of non-equilibrium thermodynamics. The Jacobi identity, representing the time structure invariance of the reversible dynamics, is found not to impose any restrictions on the convection of the morphological variables, i.e., on the divergence terms of the velocity field appearing in their evolution equations. As an application of the general compressible two-phase model, the implications of assuming both equal velocities and equal temperatures of the phases are elaborated. In particular, it is found that in such a description the volume fraction and the amount of interface per unit volume are neither scalars nor scalar densities, their intermediate convection mechanism being completely determined in terms of the material properties of the phases.