It is well-known in the Smoothed Particle Hydrodynamics (SPH) community that correction in the gradient and Laplacian operators have the potential to drastically increase the accuracy of the method at the expense of computational stability. This paper proposes a stable implementation of such corrections in all derivative operators to the Arbitrary Lagrangian Eulerian incompressible SPH (ALE-ISPH) method, in addition to a novel Neumann boundary condition (BC) applied directly on the velocity (as opposed to traditional BCs where the constraint is applied on the acceleration). In this way, the pressure is solved for both water and wall particles simultaneously, leading to a pressure field that obeys non-penetration BC and divergence-free at the same time. Furthermore, to stabilize the method, we have developed a novel density-based particle shifting technique (PST), specifically designed to deal with incompressible fluids. In this formulation, the numerical density is given as one of the most critical constraint variables. As a result, the proposed density-based PST can maintain the fluid's overall volume for the whole simulation. In addition, it also provides numerical stability as it prevents particle clustering and leads the fluid domain to an isotropic composition. First, we verified the proposed corrected formulation with the novel Neumann BC for both non-penetration and non-slip conditions with the simulation of hydrostatic pressure and Poisenuille flow, respectively. Then, we tested the proposed density-based PST with the rotating square patch problem with results comparable to previous studies. Lastly, we verified the proposed method for the dam break with an obstacle test, a highly dynamic problem.
This paper presents a computational methodology developed for a high-order approximation of compressible fluid dynamics equations with discontinuities. The methodology is based on a discontinuous Galerkin spectral-element method (DGSEM) built upon a split discretization framework with summation-by-parts (SBP) property, which mimics the integration-by-parts operation in a discrete sense. To extend the split DGSEM framework to discontinuous cases, we implement a shock capturing method based on the entropy viscosity formulation. The developed high-order split-form DGSEM with shock-capturing methodology is subject to a series of evaluation on both one-dimensional and two-dimensional, continuous and discontinuous cases. Convergence of the method is demonstrated both for smooth and shocked cases that have analytical solutions. The 2D Riemann problem tests illustrate an accurate representation of all the relevant flow phenomena, such as shocks, contact discontinuities, and rarefaction waves. All test cases are able to run with a polynomial order of 7 or higher. The values of the tunable parameters related to the entropy viscosity are robust for both 1D and 2D test problems. We also show that higher-order approximations yield smaller errors than lower-order approximations, for the same number of total degrees of freedom.
We propose a physics-driven stretching function for direct numerical simulation (DNS) of compressible turbulent wall-bounded flows, which blends uniform near-wall spacing with uniform resolution in terms of semi-local Kolmogorov units in the outer wall layer. Given target Mach number, Reynolds number and wall temperature, our procedure yields a well-defined prescription for the number of grid points and their distribution which guarantee at the same time numerical accuracy and judicious exploitation of computational resources. DNS of high-speed turbulent boundary layers are used to evaluate the quality of the proposed stretching function, which show that one can achieve identical results as with general-purpose stretching functions, however with substantially higher efficiency. A Python script is provided to facilitate implementation of the proposed grid stretching.
We present the three-dimensional version of the Elliptical Parcel-In-Cell (EPIC) method for the simulation of fluid flows and analogous continuum systems. The method represents a flow using a space-filling set of ellipsoidal parcels, which move, rotate and deform in the flow field. Additionally, parcels may carry any number of attributes, such as vorticity, density, temperature, etc, which generally evolve in time on the moving parcels. An underlying grid is used for efficiency in computing the velocity field from the interpolated vorticity field, and in obtaining parcel attribute tendencies. Mixing is enabled by permitting parcels to split when excessively deformed, and by merging very small parcels with the nearest other parcel. Several tests are provided which illustrate the behaviour of the method and demonstrate its effectiveness in modelling complex, buoyancy-driven turbulent fluid flows. The results are compared with a large eddy simulation (LES) and a direct numerical simulation (DNS) model.
Geometric conduction blocks stop cardiac electric propagation due to the shape or conductivity properties of the domain. The blocks are considered to cause many abnormal cardiac electric propagations, leading to cardiac electrophysiological pathologies, such as cardiac fibrillation and arrhythmia. Locating such multidimensional conduction blocks is challenging, particularly in a complex domain with a complex shape and strong anisotropy, such as the heart. To address this problem, we propose a novel mathematical model of the geometric conduction block using the relative acceleration adopted from space-time physics. An efficient numerical scheme for the mathematical model is also proposed to predict the unidirectional conduction block effectively, even in a complex domain. The relative acceleration in the cardiac electric propagation corresponds to the sink-source relationship between the excited (after repolarization) and excitable (before depolarization) cardiac cells, representing the geometric growth rate of the volume of metric balls. The trajectory is constructed from the wavefront of diffusion-reaction equations by aligning orthonormal basis vectors along the gradient of the action potential. Relative acceleration is computed along the propagational direction from the connection 1-form of the basis vectors. The proposed mathematical model and numerical scheme are applied to demonstrate geometric conduction blocks in two-dimensional (2D) simple curved domains with strong anisotropy.
