Consistent Smoothed Particle Hydrodynamics Research Articles

Eulerian consistent smoothed particle hydrodynamics (SPH) method for weakly compressible viscous flows applied to lid-driven cavity

Eulerian consistent smoothed particle hydrodynamics (SPH) method for weakly compressible viscous flows applied to lid-driven cavity

The impact of the initial core temperature on protostellar disc fragmentation

ABSTRACT Ground-based and satellite observations have revealed dust temperatures as low as ∼5−7 K in the centre of low-mass, pre-stellar cloud cores, where star formation takes place. However, external heating may rise the outer core temperatures up to ∼15−20 K. Such low temperatures at the centre of pre-stellar cores are a key factor to constrain the conditions that lead to the formation of gravitationally bound protostellar systems as was recently captured by highly-resolved Atacama large millimeter/submillimeter array observations. Here, we report consistent smoothed particle hydrodynamics collapse calculations of cold cores that demonstrate the formation of close protobinary systems via small-scale fragmentation of a gravitationally unstable protostellar disc. The results indicate that mean binary separations, of tens of astronomical units, are a consequence of disc fragmentation in cold pre-stellar cores. Cloud cores initially with temperatures ≤6 K and a low amplitude (a = 0.1), m = 2 density perturbation formed close protobinaries that were followed deep into the non-isothermal collapse for several orbital periods and appeared to survive as independent stellar entities. At temperatures ≥7 K disc fragmentation is no longer observed and the calculations terminate with the formation of a wide protobinary, which may occasionally be accompanied by small substellar objects emerging by fragmentation of the circumbinary disc. When the perturbation amplitude is raised to a = 0.25, disc fragmentation occurs again only in cores with initial temperatures ≤6 K. Therefore, increasing the perturbation amplitude does not necessarily imply that there will be disc fragmentation at higher core temperatures.

High-order consistent SPH with the pressure projection method in 2-D and 3-D

Mesh-free methods such a smoothed particle hydrodynamics (SPH) have advantages over mesh-based methods for flow in complex domains but attaining consistent high-order accurate solutions with the conventional form of SPH has yet to be resolved. The high-order smoothing error dominates the SPH error until increasingly fine resolutions cause the low-order discretisation error to dominate; this is related to discretising a volume integral into a summation. In this paper, a high-order consistency correction based on modified SPH (MSPH) and the modified finite particle method (FPM) is proposed for improving the order of the limiting discretisation error. The new technique is an arbitrarily high-order extension of these schemes where the complexity of the consistency correction and the required computations are reduced by using simplified versions of the smoothing kernel derivatives. Tested in Eulerian form, the proposed high-order consistent SPH technique (HOCSPH) is combined with new high-order SPH kernel functions, designed to improve the order of the SPH smoothing error, and the resulting hybrid technique is shown to converge according to the smoothing error initially before converging according to the HOCPSH error once the latter becomes dominant. The initial high-order convergence lowers the computational effort required to achieve higher accuracy with hybrid HOCSPH in comparison to HOCSPH with second-order smoothing accuracy kernel functions. However, for highly irregular distributions it is found that the use of kernels with second-order smoothing accuracy provides more consistent convergence properties. A number of flows are simulated in 2-D and 3-D using the new HOCSPH technique in combination with the pressure projection method, and the results show that the method is accurate and able to model highly complex flow patterns. Some issues with stability of the projection method related to pressure–velocity collocation are identified, and several remedies are proposed. While effective for the test cases herein, these remedies are new in this context and require further attention for generalisation in future studies.

Anisotropic dispersion with a consistent smoothed particle hydrodynamics scheme

A consistent smoothed particle hydrodynamics (SPH) approach is used to simulate the anisotropic dispersion of a solute in porous media. Consistency demands using large numbers of neighbors with increasing resolution. The method is tested against the anisotropic dispersion of a Gaussian contaminant plume. With irregularly distributed particles, the solution for isotropic dispersion converges to second-order accuracy when at sufficiently high resolution a large number of neighbors is used within the kernel support. For low to moderate anisotropy, the convergence rates are close to second-order, while for large anisotropic dispersion the solutions converge to better than first-order. For randomly distributed particles, the solutions are also better than first order independently of the degree of anisotropy. When negative concentrations arise, they are several orders of magnitude smaller than those encountered with standard SPH and comparable to those obtained with the MWSPH scheme of Avesani et al. The method is also insensitive to particle disorder and achieves an overall accuracy comparable to the MWSPH method using a much simpler approach.

An approximately consistent SPH simulation approach with variable particle resolution for engineering applications

This paper focuses on consistency issues caused by truncated kernels and irregular particle distributions in Smoothed-Particle-Hydrodynamics (SPH). The analysis starts from the kernel-based approximations inherent to the SPH method. An explicit kernel correction is proposed, which approximately achieves first-order consistency. The suggested approach is additionally combined with a Eulerian variable resolution approach, which conserves the number of particles but changes the individual particle mass in a transit regime. Validation studies featuring fully wetted and free-surface flow configurations are employed to assess the benefits of the combined approach. Results indicate benefits of the kernel correction and display merits and drawbacks of the variable resolution approach.

Numerical Modeling of Thermo-Mechanically Induced Stress in Substrates for Droplet-Based Additive Manufacturing Processes

Abstract Within the scope of additive manufacturing (AM) methods, a large number of popular fabrication techniques involve high-temperature droplets being targeted to a substrate for deposition. In such methods, an “ink” to be deposited is tailor-made to fit the desired application. Concentrated stresses are induced on the substrate in such procedures. A numerical simulation framework that can return quantitative and qualitative insights regarding the mechanical response of the substrate is proposed in this paper. A combined smoothed particle hydrodynamics (SPH)-finite element (FE) model is developed to solve the governing coupled thermo-mechanical equations, for the case of Newtonian inks. We also highlight the usage of consistent SPH formulations in order to recover first-order accuracy for the gradient and Laplacian operators. This allows one to solve the heat-equation more accurately in the presence of free-surfaces. The proposed framework is then utilized to simulate a hot droplet impacting a flat substrate.

A consistent spatially adaptive smoothed particle hydrodynamics method for fluid–structure interactions

A new consistent, spatially adaptive, smoothed particle hydrodynamics (SPH) method for fluid–structureinteractions (FSI) is presented. The method combines several attributes that have not been simultaneously satisfied by other SPH methods. Specifically, it employs the second-order consistent discretization for differential operators; it allows for resolutions spatially adapted with moving (translating and rotating) boundaries of arbitrary geometries; and, it accelerates the FSI solution as the adaptive approach leads to fewer degrees of freedom without sacrificing accuracy. The key ingredients in the method are a posteriori error estimator/distance-based criterion of adaptivity and a particle-shifting technique. The method is applied in simulating six different flows or FSI problems in two dimensions. The new method’s convergence, accuracy, and efficiency attributes are assessed by comparing the results it produces with analytical, finite element, and consistent SPH uniform high-resolution solutions as well as experimental data.

From Large-scale to Protostellar Disk Fragmentation into Close Binary Stars

Abstract Recent observations of young stellar systems with the Atacama Large Millimeter/submillimeter Array (ALMA) and the Karl G. Jansky Very Large Array are helping to cement the idea that close companion stars form via fragmentation of a gravitationally unstable disk around a protostar early in the star formation process. As the disk grows in mass, it eventually becomes gravitationally unstable and fragments, forming one or more new protostars in orbit with the first at mean separations of 100 au or even less. Here, we report direct numerical calculations down to scales as small as ∼0.1 au, using a consistent Smoothed Particle Hydrodynamics code, that show the large-scale fragmentation of a cloud core into two protostars accompanied by small-scale fragmentation of their circumstellar disks. Our results demonstrate the two dominant mechanisms of star formation, where the disk forming around a protostar (which in turn results from the large-scale fragmentation of the cloud core) undergoes eccentric (m = 1) fragmentation to produce a close binary. We generate two-dimensional emission maps and simulated ALMA 1.3 mm continuum images of the structure and fragmentation of the disks that can help explain the dynamical processes occurring within collapsing cloud cores.

Impetus: consistent SPH calculations of 3D spherical Bondi accretion on to a black hole

We present three-dimensional calculations of spherically symmetric Bondi accretion onto a stationary supermassive black hole (SMBH) of mass $10^{8}$ $M_{\odot}$ within a radial range of $0.02-10$ pc, using a modified version of the smoothed particle hydrodynamics (SPH) \gad \sp code, which ensures approximate first-order consistency (i.e., second-order accuracy) for the particle approximation. First-order consistency is restored by allowing the number of neighbours, $n_{\rm neigh}$, and the smoothing length, $h$, to vary with the total number of particles, $N$, such that the asymptotic limits $n_{\rm neigh}\to\infty$ and $h\to 0$ hold as $N\to\infty$. The ability of the method to reproduce the isothermal ($\gamma =1$) and adiabatic ($\gamma =5/3$) Bondi accretion is investigated with increased spatial resolution. In particular, for the isothermal models the numerical radial profiles closely match the Bondi solution, except near the accretor, where the density and radial velocity are slightly underestimated. However, as $n_{\rm neigh}$ is increased and $h$ is decreased, the calculations approach first-order consistency and the deviations from the Bondi solution decrease. The density and radial velocity profiles for the adiabatic models are qualitatively similar to those for the isothermal Bondi accretion. Steady-state Bondi accretion is reproduced by the highly resolved consistent models with a percent relative error of $\lesssim 1$\% for $\gamma =1$ and $\sim 9$\% for $\gamma =5/3$, with the adiabatic accretion taking longer than the isothermal case to reach steady flow. The performance of the method is assessed by comparing the results with those obtained using the standard Gadget and the Gizmo codes.

Smoothed Particle Hydrodynamics: A consistent model for interfacial multiphase fluid flow simulations

In this work, a consistent Smoothed Particle Hydrodynamics (SPH) model to deal with interfacial multiphase fluid flows simulation is proposed. A modification to the Continuum Stress Surface formulation (CSS) [1] to enhance the stability near the fluid interface is developed in the framework of the SPH method. A non-conservative first-order consistency operator is used to compute the divergence of stress surface tensor. This formulation benefits of all the advantages of the one proposed by Adami et al. [2] and, in addition, it can be applied to more than two phases fluid flow simulations. Moreover, the generalized wall boundary conditions [3] are modified in order to be well adapted to multiphase fluid flows with different density and viscosity. In order to allow the application of this technique to wall-bounded multiphase flows, a modification of generalized wall boundary conditions is presented here for using the SPH method. In this work we also present a particle redistribution strategy as an extension of the damping technique presented in [3] to smooth the initial transient phase of gravitational multiphase fluid flow simulations. Several computational tests are investigated to show the accuracy, convergence and applicability of the proposed SPH interfacial multiphase model.

A consistent multi-resolution smoothed particle hydrodynamics method

We seek to accelerate and increase the size of simulations for fluid–structure interactions (FSI) by using multiple resolutions in the spatial discretization of the equations governing the time evolution of systems displaying two-way fluid–solid coupling. To this end, we propose a multi-resolution smoothed particle hydrodynamics (SPH) approach in which subdomains of different resolutions are directly coupled without any overlap region. The second-order consistent discretization of spatial differential operators is employed to ensure the accuracy of the proposed method. As SPH particles advect with the flow, a dynamic SPH particle refinement/coarsening is employed via splitting/merging to maintain a predefined multi-resolution configuration. Particle regularity is enforced via a particle-shifting technique to ensure accuracy and stability of the Lagrangian particle-based method embraced. The convergence, accuracy, and efficiency attributes of the new method are assessed by simulating four different flows. In this process, the numerical results are compared to the analytical, finite element, and consistent SPH single-resolution solutions. We anticipate that the proposed multi-resolution method will enlarge the class of SPH-tractable FSI applications.

A SPH solver for simulating paramagnetic solid fluid interaction in the presence of an external magnetic field

The Smoothed Particle Hydrodynamics (SPH) method is extended to solve magnetostatic problems involving magnetically interacting solid bodies. In order to deal with the jump in the magnetic permeability at a fluid-solid interface, a consistent SPH scheme is utilized and a modified formulation is proposed to calculate the magnetic force density along the interface. The results of the magnetostatic solver are verified against those of the finite element method. The governing fluid flow equations are discretized using the same SPH scheme, developing an efficient method for simulating the motion of paramagnetic solid bodies in a fluid flow. The proposed algorithm is applied to a benchmark problem including a suspended paramagnetic solid body moving under the influence of a non-uniform magnetic field and the result is validated against literature. The proposed method is further verified by simulating the magneto-hydrodynamic interaction of two suspended circular cylinders. As a more complex test-case, the evolution of a suspended magnetic chain under the influence of a rotating magnetic field is also simulated. The deformation of a chain formed by a number of paramagnetic solid bodies in a shear flow is simulated. Steady state and dynamic responses of the magnetic chain are investigated under steady and oscillatory shear flows. The effects of Reynolds number, solid volume fraction, strength of the external magnetic field and the number of solid bodies forming the chain, are discussed.

Temporal Blending for Adaptive SPH

AbstractIn this paper, we introduce a fast and consistent smoothed particle hydrodynamics (SPH) technique which is suitable for convection–diffusion simulations of incompressible fluids. We apply our temporal blending technique to reduce the number of particles in the simulation while smoothly changing quantity fields. Our approach greatly reduces the error introduced in the pressure term when changing particle configurations. Compared to other methods, this enables larger integration time‐steps in the transition phase. Our implementation is fully GPU‐based to take advantage of the parallel nature of particle simulations.

Non-isothermal diffusion in a binary mixture with smoothed particle hydrodynamics

We explore the possibility of controlling the pattern formation in a purely diffusive binary mixture described by a van der Waals equation of state in non-isothermal situations. Simulations are conducted with a previously formulated thermodynamically consistent smoothed particle hydrodynamics model for a phase separating mixture. We focus on the effect of the overall concentration of species in the patterns caused by a point-like sink of energy.