High order difference schemes using the Local Anisotropic Basis Function Method

Mesh-free methods have significant potential for simulations in complex geometries, as the time consuming process of mesh-generation is avoided. Smoothed Particle Hydrodynamics (SPH) is the most widely used mesh-free method, but suffers from a lack of consistency. High order, consistent, and local (using compact computational stencils) mesh-free methods are particularly desirable. Here we present a novel framework for generating local high order difference operators for arbitrary node distributions, referred to as the Local Anisotropic Basis Function Method (LABFM). Weights are constructed from linear sums of anisotropic basis functions (ABFs), chosen to ensure exact reproduction of polynomial fields up to a given order. The ABFs are based on a fundamental Radial Basis Function (RBF), and the choice of fundamental RBF has small effect on accuracy, but influences stability. LABFM is able to generate high order difference operators with compact computational stencils (4th order with 25 nodes, 8th order with 60 nodes in two dimensions). At domain boundaries (with incomplete support) LABFM automatically provides one-sided differences of the same order as the internal scheme, up to 4th order. We use the method to solve elliptic, parabolic and mixed hyperbolic-parabolic PDEs, showing up to 8th order convergence. The inclusion of hyperviscosity is straightforward, and can effectively provide stability when solving hyperbolic problems. LABFM is a promising new mesh-free method for the numerical solution of PDEs in complex geometries. The method is highly scalable, and for Eulerian schemes, the computational efficiency is competitive with RBF-FD for a given accuracy. A particularly attractive feature is that in the low order limit, LABFM collapses to SPH, and there is potential for Arbitrary Lagrangian-Eulerian schemes with natural adaptivity of resolution and accuracy.Comment: Accepted manuscript: 28 pages, 23 figures. Accepted in J. Comput. Phys. 10th May 202

Large eddy simulations of bubbly flows and breaking waves with smoothed particle hydrodynamics

For turbulent bubbly flows, multi-phase simulations resolving both the liquid and bubbles are prohibitively expensive in the context of different natural phenomena. One example is breaking waves, where bubbles strongly influence wave impact loads, acoustic emissions and atmospheric-ocean transfer, but detailed simulations in all but the simplest settings are infeasible. An alternative approach is to resolve only large scales, and model small-scale bubbles adopting sub-resolution closures. Here, we introduce a large eddy simulation smoothed particle hydrodynamics (SPH) scheme for simulations of bubbly flows. The continuous liquid phase is resolved with a semi-implicit isothermally compressible SPH framework. This is coupled with a discrete Lagrangian bubble model. Bubbles and liquid interact via exchanges of volume and momentum, through turbulent closures, bubble breakup and entrainment, and free-surface interaction models. By representing bubbles as individual particles, they can be tracked over their lifetimes, allowing closure models for sub-resolution fluctuations, bubble deformation, breakup and free-surface interaction in integral form, accounting for the finite time scales over which these events occur. We investigate two flows: bubble plumes and breaking waves, and find close quantitative agreement with published experimental and numerical data. In particular, for plunging breaking waves, our framework accurately predicts the Hinze scale, bubble size distribution, and growth rate of the entrained bubble population. This is the first coupling of an SPH framework with a discrete bubble model, with potential for cost-effective simulations of wave–structure interactions and more accurate predictions of wave impact loads

Boundary conditions for simulations of oscillating bubbles using the non-linear acoustic approximation

AbstractWe have developed a new boundary condition for finite volume simulations of oscillating bubbles. Our method uses an approximation to the motion outside the domain, based on the solution at the domain boundary. We then use this approximation to apply boundary conditions by defining incoming characteristic waves at the domain boundary. Our boundary condition is applicable in regions where the motion is close to spherically symmetric. We have tested our method on a range of one- and two-dimensional test cases. Results show good agreement with previous studies. The method allows simulations of oscillating bubbles for long run times (5×105 time steps with a CFL number of 0.8) on highly truncated domains, in which the boundary condition may be applied within 0.1% of the maximum bubble radius. Conservation errors due to the boundary conditions are found to be of the order of 0.1% after 105 time steps. The method significantly reduces the computational cost of fixed grid finite volume simulations of oscillating bubbles. Two-dimensional results demonstrate that highly asymmetric bubble features, such as surface instabilities and the formation of jets, may be captured on a small domain using this boundary condition

The Kaye effect: new experiments and a mechanistic explanation

The Kaye effect is a phenomenon whereby a jet of fluid poured onto a surface appears to leap on impact, rather than stagnate or coil as expected. Since it was first described in 1963, several authors have attempted to explain the mechanism by which the phenomenon occurs, although to date no complete explanation for the behaviour exists. Current evidence points towards the existence of an air layer between the jet and the heap which enables slip. We show that the Kaye effect does not occur in a vacuum, indicating that the air layer is crucial for the effect to occur. By use of control volume analysis we show that viscoelasticity plays a key role in the Kaye effect, and this role is two-fold. Firstly, viscoelasticity appears to increase air entrainment, and secondly, it reduces the pressure required to bend the jet, allowing a thicker air layer to be sustained. Shear thinning behaviour reduces this viscoelastic response. These findings provide new insight into a problem that has puzzled rheologists for over half a century

High Weissenberg number simulations with incompressible Smoothed Particle Hydrodynamics and the log-conformation formulation

Viscoelastic flows occur widely, and numerical simulations of them are important for a range of industrial applications. Simulations of viscoelastic flows are more challenging than their Newtonian counterparts due to the presence of exponential gradients in polymeric stress fields, which can lead to catastrophic instabilities if not carefully handled. A key development to overcome this issue is the log-conformation formulation, which has been applied to a range of numerical methods, but not previously applied to Smoothed Particle Hydrodynamics (SPH). Here we present a 2D incompressible SPH algorithm for viscoelastic flows which, for the first time, incorporates a log-conformation formulation with an elasto-viscous stress splitting (EVSS) technique. The resulting scheme enables simulations of flows at high Weissenberg numbers (accurate up to Wi = 85 for Poiseuille flow). The method is robust, and able to handle both internal and free-surface flows, and a range of linear and non-linear constitutive models. Several test cases are considered including flow past a periodic array of cylinders and jet buckling. This work presents a significant step change in capabilities compared to previous SPH algorithms for viscoelastic flows, and has the potential to simulate a wide range of new and challenging applications

High-order simulations of isothermal flows using the local anisotropic basis function method (LABFM)

Mesh-free methods have significant potential for simulations of flows in complex geometries, with the difficulties of domain discretisation greatly reduced. However, many mesh-free methods are limited to low order accuracy. In order to compete with conventional mesh-based methods, high order accuracy is essential. The Local Anisotropic Basis Function Method (LABFM) is a mesh-free method introduced in King et al. (2020) [20], which enables the construction of highly accurate difference operators on disordered node discretisations. Here, we introduce a number of developments to LABFM, in the areas of basis function construction, stencil optimisation, stabilisation, variable resolution, and high order boundary conditions. With these developments, direct numerical simulations of the Navier Stokes equations are possible at extremely high order (up to 10th order in characteristic node spacing internally). We numerically solve the isothermal compressible Navier Stokes equations for a range of geometries: periodic and channel flows, flows past a cylinder, and porous media. Excellent agreement is seen with analytical solutions, published numerical results (using a spectral element method), and experiments. The potential of the method for direct numerical simulations in complex geometries is demonstrated with simulations of subsonic and transonic flows through an inhomogeneous porous media at pore Reynolds numbers up to Rep = 968

High order difference schemes using the local anisotropic basis function method

Mesh-free methods have significant potential for simulations in complex geometries, as the time consuming process of mesh-generation is avoided. Smoothed Particle Hydrodynamics (SPH) is the most widely used mesh-free method, but suffers from a lack of consistency. High order, consistent, and local (using compact computational stencils) mesh-free methods are particularly desirable. Here we present a novel framework for generating local high order difference operators for arbitrary node distributions, referred to as the Local Anisotropic Basis Function Method (LABFM). Weights are constructed from linear sums of anisotropic basis functions (ABFs), chosen to ensure exact reproduction of polynomial fields up to a given order. The ABFs are based on a fundamental Radial Basis Function (RBF), and the choice of fundamental RBF has small effect on accuracy, but influences stability. LABFM is able to generate high order difference operators with compact computational stencils (4th order with ≈ 25 nodes, 8th order with ≈ 60 nodes in two dimensions). At domain boundaries (with incomplete support) LABFM automatically provides one-sided differences of the same order as the internal scheme, up to 4th order. We use the method to solve elliptic, parabolic and mixed hyperbolic-parabolic partial differential equations (PDEs), showing up to 8th order convergence. The inclusion of hyperviscosity is straightforward, and can effectively provide stability when solving hyperbolic problems. LABFM is a promising new mesh-free method for the numerical solution of PDEs in complex geometries. The method is highly scalable, and for Eulerian schemes, the computational efficiency is competitive with RBF-FD for a given accuracy. A particularly attractive feature is that in the low order limit, LABFM collapses to Smoothed Particle Hydrodynamics (SPH), and there is potential for Arbitrary Lagrangian-Eulerian schemes with natural adaptivity of resolution and accuracy

High-order velocity and pressure wall boundary conditions in Eulerian incompressible SPH

High-order velocity and pressure boundary conditions are presented in Eulerian incompressible smoothed particle hydrodynamics (ISPH). While the high-order convergence of Eulerian ISPH has been demonstrated by the authors for periodic internal flows using Gaussian kernels this was limited by first to second-order accuracy for cases with solid boundaries. Since the SPH interpolation method is numerically robust there is potential for obtaining high-order accuracy in topologically complex domains with robust high-order accurate boundary conditions. In this paper high-order finite-difference extrapolation methods at solid boundaries are developed in Eulerian ISPH to allow for enforcement of the Dirichlet boundary condition for velocity and the Neumann boundary condition for pressure with high-order accuracy. Convergence up to fourth-order is demonstrated for 2-D Taylor-Couette flow and 3-D simulations of Taylor-Couette cellular flow structures are used to demonstrate accuracy and robustness. The order of accuracy may be extended to even higher-order using the analysis presented. Compact fourth-order Wendland-type kernels have also been derived to reduce the particle support region thereby lowering computational effort without loss of high-order convergence. The proposed formulation is therefore entirely high order

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