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![The simulations of the Rayleigh-Taylor instability at Re = 420; the interface position obtained with the standard SPH approach (Eqs. (18), (14), (15)); the black points denote the reference data computed with the Level-Set formulation (312 × 634 cells) [8].](profile/K-Szewc/publication/261873711/figure/fig1/AS:302517960364032@1449137310810/The-simulations-of-the-Rayleigh-Taylor-instability-at-Re-420-the-interface-position.png)
The simulations of the Rayleigh-Taylor instability at Re = 420; the interface position obtained with the standard SPH approach (Eqs. (18), (14), (15)); the black points denote the reference data computed with the Level-Set formulation (312 × 634 cells) [8].
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Smoothed Particle Hydrodynamics (SPH) is a fully Lagrangian, particle-based technique for fluid-flow computations. The main advantage over Eulerian techniques is no requirement of the grid, therefore this is a natural approach to simulate multi-phase flows. The main purpose of this study is an overview and the critical analysis of the SPH variants...
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... due to the computational efficiency, their usefulness is limited [ 10], and, therefore, those approaches are not discussed here. To verify the usefulness of the SPH formulations for incompressible multiphase flow computations, the case of the Rayleigh-Taylor instability has been simulated. The validation case involves two immiscible fluids enclosed in a rectangular domain of width L and height 2 L . Initially, the phases are separated by the interface located at y = L [1 − 0 . 15 sin (2 πx/L )]. The lower component has density L = 0 , while the upper one U = 1 . 8 0 . The kinematic viscosity is equal in both phases ( ν = ν L = ν U ). Since the system occurs under gravity g = [0 , − g ] and the upper phase is heavier, in the absence of the surface tension an instability always arises and vorticity is generated. For such a flow, the Reynolds number may be defined as Re = L 3 g/ν = 420. The simulations were performed with 120 × 240 particles initially homogeneously distributed in the domain. To compute the hydrostatic force, we use the technique described in [21], where the hydrostatic pressure is computed on a regular mesh and later projected on the particles. Figure 1 presents the SPH simulations compared to the Level-Set reference solutions (312 × 624 cells) obtained by Grenier et al. [8]. Presented data were computed at t = 1 , 3 , 5, normalized with the convective time scale T conv = L/g , and solved using Eqs. (14), (15) and (18), called here the standard formulation. Unfortunately, for the developed flow ( t = 5) the SPH results are not in good accordance with the Level-Set data. This discrepancy is caused by the use of Eq. (14). This symmetrical (with particle swapping) SPH form is constructed with the identity (13). Since the last term of Eq. (13) contains density gradient that is very high on the interface, this is the source of some instabilities ...
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Citations
... After having tested a number of benchmarks in our in-house code [1,2], we moved to an industrial application: the moon pool. This case was suggested by ESTECO SpA, the optimization software provider located in Trieste (Italy), which is a partner of the present research. ...
In maritime heavy industry, the moon pool is an opening through the hull of some vessels. We replicate a generic design of the pool with the Lagrangian approach Smoothed Particle Hydrodynamics (SPH) whose mathematical model efficiently simulates the interphasial surface. In this study we present the capability of our in-house SPH code to handle the sloshing in the moon pool. We validate the SPH approach to flows for the sloshing tank benchmark and show the stability of this approach for the non-trivial case of the moon pool.
... Szewc et al . systematically studied the SPH application in multi-phase flows and compared different surface tension models [43] . These works do display diverse ideas in surface modeling with SPH, particularly in curvature representation based on CSF; however, they are multiphase solvers, which are relatively expensive compared with the single-phase solver [44] . ...
Surface tension plays a crucial role in many practical issues ranging from the bubble and droplet dynamics to the super-hydrophilic and hydrophobic bio-inspired designs. Current smoothed particle hydrodynamics models for surface tension are mostly multiphase-based and computationally expensive. In this work, we propose a single-phase model of surface tension based on fast free-surface detections. We develop two free-surface detection algorithms: (1) a two-step detection combining the number of neighboring particles method and the arc-method, and (2) a central angle-based method, which is proved to be much more efficient than the reference counterparts. The central angle method is incorporated into the single-phase model. We verify the single-phase model through several droplet dynamics simulations including initially squared droplets, the oscillating droplet, binary head-on coalescence of droplets, and the wetting problems. Compared with the multiphase solver, this single-phase model is much simpler in the algorithm, more efficient in computation, and meanwhile retains good accuracy and robustness. Although the problems studied in this work are limited to 2D cases, the single-phase model lends itself naturally to large-scale 3D scenarios. The free-surface detection algorithms can also be applied in other meshfree methods.
... [18,19], using incompressible (ISPH) and weakly compressible (WCSPH) approaches respectively. Works on multiphase SPH developed by Ref. [20], [21], or [22] present results for the Rayleigh-Taylor problem to validate their different multiphase SPH or surface tension formulations, but these are analyzed only from a qualitative perspective in terms of free-surface evolution at certain characteristic instants in time. ...
... Combining Eqs. (21) and (17) the free surface perturbation can be related to the density perturbation as ...
In this article, the classical Rayleigh-Taylor instability is extended to situations where the fluid is completely confined, in both the vertical and horizontal directions. This article starts with the two-dimensional (2D) viscous periodic case with finite height where the effect of adding surface tension to the interface is analyzed. This problem is simulated from a dual perspective: first, the linear stability analysis obtained when the Navier-Stokes equations are linearized and regularized in terms of density and viscosity; and second, looking at the weakly compressible version of a multiphase smoothed particle hydrodynamics (WCSPH) method. The evolution and growth rates of the different fluid variables during the linear regime of the SPH simulation are compared to the computation of the eigenvalues and eigenfunctions of the viscous version of the Rayleigh-Taylor stability (VRTI) analysis with and without surface tension. The most unstable mode, which has the maximal linear growth rate obtained with both approaches, as well as other less unstable modes with more complex structures are reported. The classical horizontally periodic (VRTI) case is now adapted to the case where two additional left and right walls are included in the problem, representing the cases where a two-phase flow is confined in a accelerated tank. This 2D case where no periodic assumptions are allowed is also solved using both techniques with tanks of different sizes and a wide range of Atwood numbers. The agreement with the linear stability analysis obtained by a Lagrangian method such as multiphase WCSPH is remarkable.
... As a result, we confirmed the emergence of the characteristic mushroom structure in the linear growth regime. The mushroom structure shows its growth in the slanting direction because of wall boundary conditions; this is the same as the case for a classical fluid [69]. The time from the start to the final snapshot is approximately 7.49 ms in physical time, which corresponds to approximately 2.34 in the time scaled by the factor (g/L x ). Figure 5(b) represents the magnitude of the viscosity term calculated using Eq. ...
This paper presents a finite particle approximation of the two-fluid model for liquid ⁴ He using smoothed particle hydrodynamics (SPH). In recent years, several studies have combined the vortex filament model (VFM), which describes quantized vortices in superfluid components, with the Navier–Stokes equations, which describe the motion of normal fluids. These studies led us to assume that coupling both components of the two-fluid model instead of using the VFM to describe the superfluid component enables us to approximate the system. In this study, we formulated a new SPH model that simultaneously solves both equations of motion of the two-fluid model. We then performed a numerical simulation of the rotating liquid ⁴ He using our SPH. The results showed that the two major phenomena, the emergence of multiple independent vortices parallel to the circular axis and that of the so-called rigid-body rotation, can be reproduced by solving the two-fluid model using SPH. This finding is interesting because it was previously assumed that only a single vortex emerges when addressing similar problems without considering quantum mechanics. Our further analysis found that the emergence of multiple independent vortices can be realized by reformulating the viscosity term of the two-fluid model to conserve the angular momentum of the particles around their axes. Consequently, our model succeeded in reproducing the phenomena observed in quantum cases, even though we solve the phenomenological governing equations of liquid ⁴ He.
... For the simulation of surface tension between two fluids several approaches and variants were developed, most of them going back to [28] and [19] which are both based on the continuous surface force, introduced by [5]. A detailed comparison is performed by [37]. The challenge, when simulating surface tension for multi phase flows, is to account for high density ratios which dramatically affect the stability of the interface. ...
... In this section, the standard approaches to discretize the Navier-Stokes equations are shown. A standard multi-phase pressure term according to [9] and a viscosity term according to [37] are presented. Lastly the discretization of the PPE is shown, which is implemented according to [2]. ...
... The smoothing has a strong stabilizing effect on the surface and widens the area in which the surface normal is defined. The effect is also mentioned in [37]. After the smoothing of the normals, the curvature is calculated as the divergence of the normalized smoothed normalŝ n ...
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... As a result, we confirmed the emergence of the characteristic mushroom structure in the linear growth regime. The mushroom structure shows its growth in the slanting direction because of wall boundary conditions; this is the same as the case for a classical fluid [69]. The time from the start to the final snapshot is approximately 7.49 ms in physical time, which corresponds to approximately 2.34 in the time scaled by the factor (g/L x ). Figure 5(b) represents the magnitude of the viscosity term calculated using Eq. ...
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... In the present study we use a formulation proposed in [41]. To the best of our knowledge, this approach is best suited for modelling multiphase flows with large density ratios [42]. The pressure term in Eq. 6 will become ...
This work is dedicated to improved modelling of wetting phenomena and contact angles by means of meshless Smoothed Particle Hydrodynamics (SPH) method. For this purpose we modify the surface tension model in the neighbourhood of the triple line. For higher accuracy we also alter the general method of computing the interface normals, by applying a specific smoothing to the field of vectors, and not the phase indicator function. The necessity of proper boundary conditions in SPH is concisely discussed with multiphase systems in mind. The implemented model yields accurate results for a set of static cases, even for very low values of the imposed contact angle, both in the absence and presence of gravity. The results of the calculations of droplets moving along the wall show a great potential of the proposed approach in future applications.
... The section finishes with details about the implemented heat conduction as well as the time stepping algorithm. Details of the implementation can be found in Colagrossi and Landrini [2003], Szewc et al. [2012], Hirschler et al. [2017], Cleary [1998] and Crespo et al. ...
... For the calculation of the viscous forces several formulations are presented in the literature, where most of them are based on the second order derivative from equation (27). In this work the variant according to Szewc et al. [2012] for the viscosity is preferred, ...
... e.g. Szewc et al. [2012]) with known analytical curvatures (2D: 8862 m −1 , 3D: 8060 m −1 ), calculated with the novel curvature calculation in comparison with the standard approach of taking the divergence of the normalized, smoothed normals (see e.g. Hirschler et al. [2017]). ...
In Selective Laser Melting (SLM), three dimensional parts are build up from a metal powder bed by layer-wise melting with a laser. While SLM offers a high flexibility in geometrical design and material texture, the interplay of dozens of parameters is difficult to analyze experimentally. Current research focuses on numerical simulation to increase confidence in manufactured parts and to reduce the time to market. Using a Smoothed Particle Hydrodynamics (SPH) implementation on GPU, the 3D melt pool dynamics in a single laser track can be simulated within less than two hours with a reasonable spatial resolution of 3 µm. This is an extreme speed up compared to concurrent CFD methods that have recently been applied to the same problem. Using a Smoothed Particle Hydrodynamics (SPH) implementation on GPU, the 3D melt pool dynamics in a single laser track can be simulated within less than an hour with a reasonable spatial resolution of 3 µm. This is an extreme speed up compared to concurrent CFD methods that have recently been applied to the same problem. The computational efficiency allows to conduct parameter studies, that before were prohibitively expensive, to generate a virtual process. To make the simulations possible a curvature calculation technique was applied, which is novel in the field of SPH. Additionally the advantage of a direct description of the internal energy is outlined by comparing the approach with the commonly applied apparent heat approach. Designers may profit from the fast simulations in two different scenarios. Firstly, through a direct numerical simulation of certain parameter combinations and scenarios. Secondly, by informing advanced surrogate models or part scale models to predict the behavior of the entire part being produced. When adapting the heat source model, the SPH framework may also be used to describe the related Electron Beam Melting (EBM) process.
... In the present study, we use a formulation proposed by Hu and Adams (2006). To the best of our knowledge, their approach is well suited for modelling multiphase flows with large density ratios (Szewc et al., 2012b). The pressure term in Eq. (2.6) will become ...
The break-up of liquid ligaments and formation of droplets are elementary phenomena in multiphase flows which are of high importance in industrial and medical applications. From the numerical point of view, they require proper interface and surface tension treatment. In the present work, we apply Smoothed Particle Hydrodynamics, a meshless approach, to simulate the break-up of a liquid cylinder inside the gaseous phase, i.e. the Rayleigh-Plateau instability. Results obtained in 3D show that even a relatively coarse resolution allows one to predict correctly the size of droplets formed in the process. The detailed analysis of the break-up time in 2D setup implies that a certain level of spatial discretisation needs to be reached to determine this moment precisely. © 2018 Polish Society of Theoretical and Allied Mechanics. All Rights Reserved.
... The whole set of governing equations should now be expressed in the SPH formalism. However, it is important to note that it can be done in many ways [7]. For the purposes of this work, we decided to use the Collagrossi and Landrini [8] formulation. ...
