Ghost Fluid Method Research Articles

On the boundary conditions for GFMxP high-order schemes on staggered grids in the simulation of incompressible multiphase flows

The simulation of incompressible multiphase flows through the so-called fractional step method needs to solve a variable coefficient Poisson equation for discontinuous functions. Recently, it has been shown how the solution of this equation may be found out through a novel coding of the Ghost Fluid Method (named GFMxP), by avoiding any fit to evaluate the interface position and providing, anyhow, a perfect sharp modeling of the same interface. Furthermore, the accuracy order of the numerical solutions exactly corresponds to the order of the adopted finite difference scheme. The effectiveness and reliability of the new procedure were successfully checked by a lot of tests. However, the a-priori knowledge of the unknown function allowed to elude a fundamental aspect of the numerical approach: the appropriate encoding of the boundary conditions. This topic has often been debated in the past, especially from a theoretical viewpoint, and still represents a rather thorny point in the whole simulation process. The paper shows how to handle the problem in practice and in the context of the GFMxP approach, i.e. by accounting for the presence of the discontinuity and the possible use of high-order solving schemes on a staggered grid.

The GFMxP and the basic extrapolation of the ghost values to solve the Poisson equation for discontinuous functions

In a recent paper, a novel coding of the Ghost Fluid Method for the variable coefficient Poisson equation with discontinuous functions (named GFMxP) was proposed. A lot of numerical tests, with all the required quantities available in a analytic form, were used to demonstrate the ability of the new procedure in modeling a sharp interface and to check the accuracy order of the solutions. In practical applications, however, the real difficulty stands in the estimation of the so-called “ghost values”, that is the values at points where the function is not only unknown, but even not defined. These values allow to compute the corrective terms enabling the use of standard finite difference formulas in presence of a singularity and/or a discontinuity, and can be only determined through some extrapolation procedure, whose truthfulness is essential to achieve a reliable result. The paper deals with such a basic issue, by testing different numerical strategies and demonstrating the strict relationship between the order of the adopted fit-model, the order of the solving scheme for the Poisson equation and the accuracy of the final solution.

A method to represent the strain hardening effect in the hyper-elastic model within a fully Eulerian framework

We propose a method to treat the strain hardening effect in the hyperelastic model. Coupled with the level set interface capturing method and a real ghost fluid method, it is formulated in a fully Eulerian framework. The HLLD approximate Riemann solver is used to obtain the interface state. We use the elastic inverse deformation gradient tensor to track the material deformation state. Once the solid starts to yield, along the normal direction in stress space, an iterative algorithm is used to relax the elastic inverse deformation gradient tensor to target value where the equivalent stress meets the requirement of consistency condition in plasticity. This approach is intuitive and can avoid the use of the physical delay time, which is difficult to calibrate from experiments. We use the Wilkins flying plate impact, cylindrical shell collapse, Taylor impact and the conical-nosed projectile penetration test cases to validate the method. The results show good agreement with the theoretical analysis and experiments. This approach can also apply without further extension when thermal softening effect or damage models are considered.

An Edge-based Interface Tracking (EBIT) method for multiphase flows with phase change

In this paper, the Edge-Based Interface Tracking (EBIT) method is extended to simulate multiphase flows with phase change. As a novel Front-Tracking method, the EBIT method binds interfacial markers to the Eulerian grid to achieve automatic parallelization. To include phase change effects, the energy equation for each phase is solved, with the temperature boundary condition at the interface sharply imposed. When using collocated grids, the cell-centered velocity is approximately projected. This will lead to unphysical oscillations in the presence of phase change, as the velocity will be discontinuous across the interface. It is demonstrated that this issue can be addressed by using the ghost fluid method, in which the ghost velocity is set according to the jump condition, thereby removing the discontinuity. A series of benchmark tests are performed to validate the present method. It is shown that the numerical results agree very well with the theoretical solutions and the experimental results.

Turbulent natural convection in an air–water system with evaporation across the free surface

In this paper we report on direct numerical simulations of turbulent convection in a cuboidal pool that contains liquid water in the lower part and air in the upper one. The bottom wall is uniformly heated and natural convection is established in both phases, accompanied by water evaporation across the free surface of the water. The descent of the free surface due to evaporation is computed via a newly developed tracking algorithm based on the ghost-fluid method. We present results for three different cases which correspond to different pool heights. In all of them, the natural convection in the water lies in the soft-turbulence regime. Whereas in the gas, it lies in the laminar, transitional and soft-turbulence regimes, respectively. Our analysis focuses on the characteristics of the convective patterns in the two phases and the statistics of the various flow quantities of interest. According to our simulations, the flow in the water is organized in a single-roll Large-Scale Circulation (LSC). In the gas, it is organized in single or dual-roll LSCs, depending on the aspect ratio of the pool. Interestingly, the impingement points of the LSCs of the two phases at the free surface remain very close to one another, which is attributed to the continuity of the shear stresses across the free surface. Further, after the initial transient period, both the free-surface temperature and the evaporative mass flux are stabilized and remain almost constant, but they exhibit small-scale fluctuations in time due to turbulence. Also, the transport of water vapor in air has similar properties as the heat transport, and the ratio between the Sherwood and Nusselt numbers is very close to the Lewis number for air.

Pressure boundary conditions for immersed-boundary methods

Immersed boundary methods have seen an enormous increase in popularity over the past two decades, especially for problems involving complex moving/deforming boundaries. In most cases, the boundary conditions on the immersed body are enforced via forcing functions in the momentum equations, which in the case of fractional step methods may be problematic due to: i) creation of slip-errors resulting from the lack of explicitly enforcing boundary conditions on the (pseudo-)pressure on the immersed body; ii) coupling of the solution in the fluid and solid domains via the Poisson equation. Examples of fractional-step formulations that simultaneously enforce velocity and pressure boundary conditions have also been developed, but in most cases the standard Poisson equation is replaced by a more complex system which requires expensive iterative solvers. In this work we propose a new formulation to enforce appropriate boundary conditions on the pseudo-pressure as part of a fractional-step approach. The overall treatment is inspired by the ghost-fluid method typically utilized in two-phase flows. The main advantage of the algorithm is that a standard Poisson equation is solved, with all the modifications needed to enforce the boundary conditions being incorporated within the right-hand side. As a result, fast solvers based on trigonometric transformations can be utilized. We demonstrate the accuracy and robustness of the formulation for a series of problems with increasing complexity.

Coupled dynamic and thermal simulation of airfoil flap movement in turbulent flow using the lattice Boltzmann method

The study introduces an innovative approach that combines dynamic and thermal lattice Boltzmann simulations utilizing the ghost fluid boundary detection method for enhanced flow stability during flapping oscillation. This novel methodology is applied to large-eddy simulations of a flapping airfoil, aiming to capture the complex dynamics of oscillatory vortices and their impact on aerodynamics, while also allowing control over aerodynamic responses through airfoil surface temperature modulation. The investigation involves integrating the ghost fluid method into the lattice Boltzmann framework, synchronized with each time step to accurately model both the airfoil's oscillatory and boundary-induced movements. The advancement of specific fluid domain nodes due to boundary motion is managed using a recharging and force imposition scheme, estimating energy, and density function distributions on these nodes. Variations in lift and drag coefficients resulting from dynamic viscosity changes and thermal effects are observed due to airfoil surface temperature adjustments. A nonreflecting boundary condition is introduced to regulate flow velocity upper and lower near the domain boundary, facilitating smooth flow transfer from the boundary to the oscillating airfoil flow and minimizing transverse interference. Changes in energy exchange lead to delayed boundary layer separation, suggesting enhanced performance with reduced airfoil temperature. It is shown that decreasing the airfoil temperature by 100 K compared to the fluid temperature reduces the drag coefficient by 34% and increases the lift coefficient by 14%, while with an increase in 100°, the drag coefficient increases by 14.9% and the lift coefficient decreases by 4%. The proposed approach offers computational simplicity, concise solvable equations, and high accuracy, eliminating the need for mesh size adjustments when simulating different Reynolds numbers. Additionally, its ability to accommodate heat transfer-induced alterations within the aerodynamic context is highlighted. Comparative analysis with the finite volume method validates its effectiveness, demonstrating potential applications for controlling aerodynamic coefficients through controlled thermal interventions. In conclusion, the study presents a comprehensive methodology that integrates dynamic and thermal lattice Boltzmann simulations with the ghost fluid boundary detection method for enhanced flow stability during flapping oscillation. The insights gained contribute to a deeper understanding of complex aerodynamic phenomena, with implications for aerospace and fluid dynamics research, where accurate prediction and control of airfoil behavior are crucial. In summary, this study offers a groundbreaking strategy that seamlessly integrates dynamic and thermal lattice Boltzmann simulations, leveraging the ghost fluid boundary detection method for enhanced stability in flapping oscillatory motion. The outcomes contribute to a deeper understanding of intricate aerodynamic phenomena, thus holding promise for broader applications in aerospace and fluid dynamics research.

GFMxP: A novel Ghost Fluid Method to solve the variable coefficient Poisson equation

The solution of a variable coefficient Poisson equation, where the same coefficient, the unknown function and its spatial derivatives may be discontinuous, is an interesting and challenging problem, whose solution plays a fundamental role in the simulation of many physical phenomena and, in particular, of an incompressible multiphase flow. On the other hand, the Ghost Fluid Method (GFM) represents an effective numerical technique to manage derivative terms in presence of an interface (that is, a discontinuity) and to preserve its own sharp nature. Then, the method is a natural candidate to deal with the above mentioned problem. The equation, however, may be characterized by a significant jump of the coefficient across the interface, a condition which still leaves room for open questions and in which the original GFM-based approach seems not to work properly. The paper provides a comprehensive answer to these questions and proposes a novel, simpler implementation of the method (named GFMxP), testing its efficiency and robustness by a lot of well-known numerical examples.

A generalized variational level set method without frequent reinitialization for simulations of gas-liquid flows

A generalized variational level set method without frequent reinitialization process is proposed to address the issue of the traditional level set method for detailed numerical simulations of gas-liquid flows. The gas-liquid interface is captured by the level set method and the discontinuity across the interface is discretized by the ghost fluid method. A penalty term is added in the classical level set advection equation to reduce the frequent reinitialization process. The effect of the frequency of the reinitialization step on the performance of the level set method is evaluated and the effect of the penalty factor on the interface evolution is investigated in the present work. Three typical benchmark cases, including rigid body rotation, two-dimensional shear motion, and three-dimensional shear motion, are used to validate the accuracy of the present variational level set method. It is found that the reinitialization process is not necessary for every time step with the penalty term. A spatially averaged parameter near the interface is proposed to quantitatively evaluate the necessity of reinitialization. The present method is also shown to have second-order accuracy. This work provides an accurate and efficient way to simulate the gas-liquid flows.

Harten-Lax-van Leer-discontinuities with elastic waves (HLLD-e) approximate Riemann solver for two-dimensional elastic-plastic flows with slip/no-slip interface boundary conditions

We present a multi-material Harten-Lax-van Leer-discontinuities with elastic waves (HLLD-e) approximate Riemann solver with a hypoelastic model for solving two-dimensional elastic-plastic flows under tangential slip/no-slip interface boundary conditions. Shear waves and their effects have not been fully explored in HLL-type (Harten, Lax, and van Leer) Riemann solvers. To this end, transformed equations corresponding to the deviatoric stress and their Rankine–Hugoniot (RH) relations are obtained. In addition, normal interface and slip/no-slip boundary conditions are imposed separately to close the solver system. Associated with the modified ghost fluid method (MGFM), the propagation and configurations of the shear waves are examined, and the results agree with those of previous studies. In addition, rarefaction waves and fluid-structure interactions can be computed owing to the wide applicability of this solver. In the future, to better describe the behavior of plastic waves, elastoplastic results can be considered for an approximate Riemann solver.

A generalized coupled level set/volume-of-fluid/ghost fluid method for detailed simulation of gas-liquid flows

A generalized coupled level set/volume-of-fluid/ghost fluid method for the detailed simulation of gas-liquid flows is proposed in this paper. The interface is captured by the coupled level set and volume-of-fluid method, where the level set method is used to compute normal vector and curvature accurately, and the volume-of-fluid method is used to conserve the global volume or mass. The ghost fluid method is also used for discretization across the interface. The novelty of the present approach lies in that a generalized formulation for the conversion between the level set function and the volume-of-fluid function is proposed in a simple and efficient manner. Three benchmarks, including Zalesak's disk, two-dimensional deformation and three-dimensional deformation, are used for validation and verification. It shows that excellent agreement is achieved, and the present method has first-order accuracy. The current method is further applied to the simulations of droplet dynamics, rising bubbles, head-on droplet collision and liquid jet breakup, demonstrating the capability to accurately capture the interface.

On ghost fluid method-based sharp interface level set method on a co-located grid and its comparison with balanced force-based diffuse interface method

This paper, on Computational multi-Fluid Dynamics (CmFD) development, presents an extension of the Ghost Fluid Method (GFM)-based Sharp Interface Level Set Method (SI-LSM) (originally proposed on a staggered grid) for a co-located grid system. Further, this paper presents a comparative study for the GFM-based SI-LSM and a balanced force method (BFM)-based Diffuse Interface Level Set Method (DI-LSM). The BFM-based DI-LSM avoids a pressure-interfacial force decoupling, during the implementation of the MIM, by a proper-discretization of the surface tension term; not needed for the single-phase flow. Whereas, the GFM-based SI-LSM implicitly couples two immiscible, incompressible fluids via interface jump condition, which makes the Momentum Interpolation Method (MIM) for the two-fluid flow same as that for a single-fluid flow. A unified, for both SI-LSM and DI-LSM, mathematical and numerical formulations are presented. Order of accuracy is demonstrated in-between first-order and second-order; similar to that reported earlier for the various CmFD methods. Finally, validation and relative-performance study are presented for four test cases: static droplet, dam break, drop coalescence, and Rayleigh-Taylor instability. For the static droplet, the relative-performance of the present SI-LSM as compared to the DI-LSM is found similar to that in the literature for SI-VOF (Volume of Fluid) method as compared to DI-VOF method. For the various test cases, superior performance is demonstrated for the SI-LSM as compared to DI-LSM. This paper presents a broader perspective to the MIM for a single-versus-two fluid-flow and diffuse-versus-sharp approaches; and a novel GFM-based SI-LSM on a co-located grid.

An efficient tracking method of evaporative and flat free surfaces for turbulent convection

In this paper, we present a tracking algorithm of evaporative and flat free surfaces on collocated grids. This algorithm combines the ghost-fluid method with a flux-interpolation technique and is well suited for the numerical treatment of gas–liquid flows involving turbulent convection, mixing, and phase change via evaporation across a free surface. Further, it can accurately take into account the jumps in the values of the various flow quantities across the free surface and the time-dependent interaction between the two phases. The efficiency of the proposed algorithm is illustrated via numerical tests. The first test involves Couette flow above a liquid at rest; our numerical predictions for the evaporative mass flux compares very favorably with the analytical solution. The second test involves 2D simulations and grid-convergence study of free convection with evaporation in a system comprising a column of water and a layer of air above it. Cases at different Rayleigh numbers are considered and our numerical predictions are compared with experimental data and with earlier single-phase simulations based on approximate evaporation models. Further, we present and discuss results from 3D numerical simulations involving evaporation across the free surface, turbulent convection below it (liquid side) and free convection with mixing above it (gas side). Our numerical results are in good agreement with experiments and show that the algorithm provides accurate predictions of the evaporative mass flux and is robust enough to handle the descent of the free surface and simultaneously resolve the emerging turbulent structures.

集束超音波中での気泡クラウド形成に関するマルチスケールシミュレーション

This study deals with a numerical simulation of the growth of bubble nuclei and the corresponding bubble cloud formation in a pressure field given by high intensity focused ultrasound (HIFU) backscattered from a laser-induced bubble interface. In the present paper, using a multiscale numerical method (Tamura et al., Japanese J. Multiphase Flow, 2022) in which the ghost fluid method is coupled with bubble dynamics, the growth of the bubble cloud which consists of multiple bubble layers is simulated continuously as observed in the experiments. It is shown that the distance between the first cavitation inception point and a laser-induced bubble interface is about 0.27λwhere λ is the wavelength of HIFU; the distance between bubble layers in the cloud is also about 0.35λ these distances in the simulation are in good agreement with the experiments. The influence of the configuration of bubble nuclei on the final form of a cone-shaped bubble cloud is investigated. It is also shown that the shape of each bubble layer, which depends on the initial configuration of bubble nuclei, affects the final formation of bubble clouds. This is because that the negative pressure due to the backscattering of HIFU is dependent on the shape of each bubble layer. The results also show that the averaged final shape of bubble clouds quantitively matches the results as observed in the experiment by Horiba et al. (J. Acoust. Soc. Am., 2020).

Adaptive interface-Mesh un-Refinement (AiMuR) based sharp-interface level-set-method for two-phase flow

Adaptive interface-Mesh un-Refinement (AiMuR) based Sharp-Interface Level-Set-Method (SI-LSM) is proposed for both uniform and non-uniform Cartesian-Grid. The AiMuR involves interface location based dynamic un-refinement (with merging of the four control volumes) of the Cartesian grid away from the interface. The un-refinement is proposed for the interface solver only. A detailed numerical methodology is presented for the AiMuR and ghost-fluid method based SI-LSM. Advantage of the novel as compared to the traditional SI-LSM is demonstrated with a detailed qualitative as well as quantitative performance study, involving the SI-LSMs on both coarse grid and fine grid, for three sufficiently different two-phase flow problems: dam break, breakup of a liquid jet and drop coalescence. A superior performance of AiMuR based SI-LSM is demonstrated - the AiMuR on a coarser non-uniform grid ( $$NU_{c}^{AiMuR}$$ ) is almost as accurate as the traditional SI-LSM on a uniform fine grid ( $$U_{f}$$ ) and takes a computational time almost same as that by the traditional SI-LSM on a uniform coarse grid ( $$U_{c}$$ ). The AMuR is different from the existing Adaptive Mesh Refinement (AMR) as the former involves only mesh un-refinement while the later involves both refinement and un-refinement of the mesh. Moreover, the proposed computational development is significant since the present adaptive un-refinement strategy is much simpler to implement as compared to that for the commonly used adaptive refinement strategies. The proposed numerical development can be extended to various other multi-physics, multi-disciplinary and multi-scale problems involving interfaces.

PERFORMANCE OF SHARP-VERSUS-DIFFUSE INTERFACE-BASED LEVEL SET METHOD ON A STAGGERED-VERSUS-CO-LOCATED GRID FOR CMFD

The present work is on comparison of computational performance for various level set methods (LSMs): diffuse interface level set method on staggered grid (DI-LSM<sub>stag</sub>), sharp-interface level set method on staggered grid (SI-LSM<sub>stag</sub>), diffuse interface level set method on co-located grid (DI-LSM<sub>col</sub>), and sharp interface level set method on co-located grid (SI-LSM<sub>col</sub>). Even though the implementations of the diffuse and sharp interface (DI and SI) approaches on staggered grid are straightforward, an additional pressure-interfacial force balance needs to be ensured on the co-located grid. This is established here with balanced force method (BFM) for the DI-LSM and ghost fluid method (GFM) for the SI-LSM. Computational performances of these LSMs are presented for a variety of computational multi-fluid dynamics (CMFD) problems: static drop, dam break, rising bubble, falling droplet, and droplet coalescence. Greater accuracy is found with SI-LSMs for the static drop, dam break, and rising bubble, whereas for the other problems, both SI-LSM and DI-LSM result in almost the same accuracy. Smaller computational time is taken by the SI-LSM for rising bubble and falling droplet, and by DI-LSM for the dam break and droplet coalescence. Comparing between grid systems, co-located grid resulted in greater accuracy for all the problems except falling droplet, for which both grid systems resulted in similar accuracy, whereas, a smaller computational time is taken by the co-located grid for rising bubble and falling droplet, and by the staggered grid for dam break and droplet coalescence. Overall, SI-LSM on the co-located grid shows better results with a slight increase in computational time as compared to the other LSMs, and is a suitable alternative to the staggered grid.

NUMERICAL INVESTIGATION OF BUBBLE CLOUD FORMATION BY HIGH-INTENSITY FOCUSED ULTRASOUND USING THE GHOST FLUID METHOD COUPLED WITH BUBBLE DYNAMICS

In this study, numerical simulations were conducted to investigate the growth of bubble nuclei and the corresponding bubble cloud formation in a pressure field given by high-intensity focused ultrasound (HIFU) backscattered from the bubble interface. A multiscale numerical method was developed to simulate the corresponding experiment, in which the overall pressure fields obtained by HIFU were solved using the ghost fluid method coupled with the bubble dynamics in the microscopic field. The validity of the present method was confirmed by the growth of a single bubble nucleus placed on the HIFU propagation axis. The formation of a cone-shaped bubble cloud that consisted of multiple layers of bubbles observed in the experiment was simulated successfully under the random distribution of initial bubble nuclei. The influence of the initial distribution of bubble nuclei on the cone-shaped bubble cloud formation was also investigated. The results showed that the shape of each bubble layer affects the final form of the bubble cloud since the negative pressure due to backscattering from the HIFU is dependent on the shape of each bubble layer. A final bubble cloud shape was predicted by averaging the void fractions in all of the bubble nuclei distribution cases. The results showed that the averaged final shape of the bubble clouds matched the results observed in the experiment: the distances between the first cavitation inception point and the laser-induced bubble interface and between the bubble layers in the cloud were in quantitative agreement with the experiments.

Lee model based numerical scheme for steady vapor chamber simulations

Vapor chamber is a phase-change based heat spreader widely used in thermal management. Steady numerical simulation is a useful practice for vapor chamber design and optimization. In numerical simulations, assuming incompressible flow and isothermal vapor core is common to alleviate the stability issues raised at liquid-vapor interfaces because the mass/energy transfers at such interfaces are sensitive to the fluctuations of pressure, temperature, and density. It is also typical to preset a single kind of phase change behavior at a liquid-vapor interface (either condensation or evaporation). However, these assumptions could no longer be appropriate when input heat flux is high, the ratio of heating area to evaporator area is small, and/or pillars are presented. This paper presents a novel numerical scheme for vapor chamber simulations by accounting for both the flow compressibility and the temperature variations within the vapor core while applying mass/energy source terms to liquid-vapor interfaces based on the Lee model through the ghost fluid method. No preset of the phase change behavior at liquid-vapor interfaces is needed as well. The results show that, under the input heat flux from 5.6 to 38.9 W/cm2, the average temperature errors between the experiments and the simulations range from 3.0% to 5.0% for the heating area, 5.5% to 8.0% for the evaporator, and 1.4% to 3.0% for the condenser. Although these temperature errors are not exceptional, this proposed scheme is numerically stable by exhibiting good grid independence and being able to converge in 128 iterations over four million elements. Furthermore, attributed to fewer assumptions needed, this proposed scheme can simulate flow characteristics that well agree with the sense of phase change behaviors and is also applicable to other two-phase based heat spreaders (such as heat pipes) at various conditions in structure, dimension, heat input, and convergence criteria.

Riemann solvers for phase transition in a compressible sharp-interface method

In this paper, we consider Riemann solvers with phase transition effects based on the Euler–Fourier equation system. One exact and two approximate solutions of the two-phase Riemann problem are obtained by modelling the phase transition process via the theory of classical irreversible thermodynamics. Closure is obtained by appropriate Onsager coefficients for evaporation and condensation. We use the proposed Riemann solvers in a sharp-interface level-set ghost fluid method to couple the individual phases with each other. The proposed sharp-interface method is validated against molecular dynamics data of evaporating Lennard–Jones truncated and shifted fluid. We further study the effects of phase transition on a shock-drop interaction with the novel approximate Riemann solvers.

A GRP-based high resolution ghost fluid method for compressible multi-medium fluid flows I: One-dimensional case

This paper proposes a generalized Riemann problem (GRP)-based high resolution ghost fluid method (GFM) for the simulation of 1-D multi-medium compressible fluid flows. A kind of linearly distributed ghost fluid states is defined via a local double-medium generalized Riemann problem (GRP) at the material interface. The advantages of the GRP-based GFM over the RP-based GFM (MGFM) are reflected in the following aspects: (i) The GRP-based GFM can maintain the continuity of the material derivatives of the pressure across material interfaces, so that the pressure mismatch in the RP-based GFM (MGFM) can be eliminated dramatically, even for long time computation. (ii) The initial data for the associated Riemann problem of the local double-medium GRP are second-order approximation for the fluid states at material interfaces, and the initial data for the local double-medium GRP are also second-order approximation for the fluid states near material interfaces, so that the numerical accuracy are increased greatly. (iii) The GRP-based GFM can reflect the thermodynamical properties of different mediums, which have fundamental importance for the study of compressible fluid flow, so that the overheating errors in the RP-based GFM (MGFM) can be suppressed. Several typical numerical examples display the excellent performance of our new method.