Xiangyu Hu

A multi-phase SPH method for macroscopic and mesoscopic flows

A multi-phase smoothed particle hydrodynamics (SPH) method for both macroscopic and mesoscopic flows is proposed. Since the particle-averaged spatial derivative approximations are derived from a particle smoothing function in which the neighboring particles only contribute to the specific volume, while maintaining mass conservation, the new method handles density discontinuities across phase interfaces naturally. Accordingly, several aspects of multi-phase interactions are addressed. First, the newly formulated viscous terms allow for a discontinuous viscosity and ensure continuity of velocity and shear stress across the phase interface. Based on this formulation thermal fluctuations are introduced in a straightforward way. Second, a new simple algorithm capable for three or more immiscible phases is developed. Mesocopic interface slippage is included based on the apparent slip assumption which ensures continuity at the phase interface. To show the validity of the present method numerical examples on capillary waves, three-phase interactions, drop deformation in a shear flow, and mesoscopic channel flows are considered.

An incompressible multi-phase SPH method

An incompressible multi-phase SPH method is proposed. In this method, a fractional time-step method is introduced to enforce both the zero-density-variation condition and the velocity-divergence-free condition at each full time-step. To obtain sharp density and viscosity discontinuities in an incompressible multi-phase flow a new multi-phase projection formulation, in which the discretized gradient and divergence operators do not require a differentiable density or viscosity field is proposed. Numerical examples for Taylor-Green flow, capillary waves, drop deformation in shear flows and for Rayleigh-Taylor instability are presented and compared to theoretical solutions or references from literature. The results suggest good accuracy and convergence properties of the proposed method.

A generalized wall boundary condition for smoothed particle hydrodynamics

In this paper we present a new formulation of the boundary condition at static and moving solid walls in SPH simulations. Our general approach is both applicable to two and three dimensions and is very simple compared to previous wall boundary formulations. Based on a local force balance between wall and fluid particles we apply a pressure boundary condition on the solid particles to prevent wall penetration. This method can handle sharp corners and complex geometries as is demonstrated with several examples. A validation shows that we recover hydrostatic equilibrium conditions in a static tank, and a comparison of the classical dam break simulation with state-of-the-art results in literature shows good agreement. We simulate various problems such as the flow around a cylinder and the backward facing step at Re=100 to demonstrate the general applicability of this new method.

A conservative interface method for compressible flows

In this work, we present a conservative interface method for both multi-fluid and complex boundary problems, in which the standard finite volume scheme on Cartesian grids is modified by considering computational cells being cut by interface. While the discretized governing equations are updated conservatively, the method treats the topological changes naturally by combining interface description and geometric operations with a level set technique. Extensive tests in 1D are carried out, and 2D examples suggest that the present scheme is able to handle multi-fluid and complex (static or moving) boundary problems in a straightforward way with good robustness and accuracy.

An adaptive central-upwind weighted essentially non-oscillatory scheme

In this work, an adaptive central-upwind 6th-order weighted essentially non-oscillatory (WENO) scheme is developed. The scheme adapts between central and upwind schemes smoothly by a new weighting relation based on blending the smoothness indicators of the optimal higher order stencil and the lower order upwind stencils. The scheme achieves 6th-order accuracy in smooth regions of the solution by introducing a new reference smoothness indicator. A number of numerical examples suggest that the present scheme, while preserving the good shock-capturing properties of the classical WENO schemes, achieves very small numerical dissipation.

A new surface-tension formulation for multi-phase SPH using a reproducing divergence approximation

In this paper, we propose a new surface-tension formulation for multi-phase smoothed particle hydrodynamics (SPH). To obtain a stable and accurate scheme for surface curvature, a new reproducing divergence approximation without the need for a matrix inversion is derived. Furthermore, we introduce a density-weighted color-gradient formulation to reflect the reality of an asymmetrically distributed surface-tension force. We validate our method with analytic solutions and demonstrate convergence for different cases. Furthermore, we show that our formulation can handle phase interfaces with density and viscosity ratios of up to 1000 and 100, respectively. Finally, complex three-dimensional simulations including breakup of an interface demonstrate the capabilities of our method.

A constant-density approach for incompressible multi-phase SPH

A constant-density approach, which corrects intermediate density errors by adjusting the half-time-step velocity with exact projection, is proposed for the multi-phase SPH method developed in our previous work [X.Y. Hu, N.A. Adams, An incompressible multi-phase SPH method, J. Comput. Phys. 227 (2007) 264-278]. As no prescribed reference pressure is required, the present approach introduces smaller numerical viscosity and allows to simulate flows with unprecedentedly high density ratios by the projection SPH method. Numerical examples for Taylor-Green flow, capillary waves and for Rayleigh-Taylor instability are presented and compared to theoretical solutions or references from the literature.

Positivity-preserving method for high-order conservative schemes solving compressible Euler equations

In this work a simple method to enforce the positivity-preserving property for general high-order conservative schemes is proposed for solving compressible Euler equations. The method detects critical numerical fluxes which may lead to negative density and pressure, and for such critical fluxes imposes a simple flux limiter by combining the high-order numerical flux with the first-order Lax-Friedrichs flux to satisfy a sufficient condition for preserving positivity. Though an extra time-step size condition is required to maintain the formal order of accuracy, it is less restrictive than those in previous works. A number of numerical examples suggest that this method, when applied on an essentially non-oscillatory scheme, can be used to prevent positivity failure when the flow involves vacuum or near vacuum and very strong discontinuities.In this work a simple method to enforce the positivity-preserving property for general high-order conservative schemes is proposed. The method keeps the original scheme unchanged and detects critical numerical fluxes which may lead to negative density and pressure, and then imposes a cut-off flux limiter to satisfy a sufficient condition for preserving positivity. Though an extra time-step size condition is required to maintain the formal order of accuracy, it is less restrictive than those in previous works. A number of numerical examples suggest that this method, when applied on an essentially non-oscillatory base scheme, can be used to prevent positivity failure when the flow involves vacuum or near vacuum and very strong discontinuities.

A conservative immersed interface method for Large-Eddy Simulation of incompressible flows

We propose a conservative, second-order accurate immersed interface method for representing incompressible fluid flows over complex three dimensional solid obstacles on a staggered Cartesian grid. The method is based on a finite-volume discretization of the incompressible Navier-Stokes equations which is modified locally in cells that are cut by the interface in such a way that accuracy and conservativity are maintained. A level-set technique is used for description and tracking of the interface geometry, so that an extension of the method to moving boundaries and flexible walls is straightforward. Numerical stability is ensured for small cells by a conservative mixing procedure. Discrete conservation and sharp representation of the fluid-solid interface render the method particularly suitable for Large-Eddy Simulations of high-Reynolds number flows. Accuracy, second-order grid convergence and robustness of the method is demonstrated for several test cases: inclined channel flow at Re=20, flow over a square cylinder at Re=100, flow over a circular cylinder at Re=40, Re=100 and Re=3900, as well as turbulent channel flow with periodic constrictions at Re=10,595.

On the HLLC Riemann solver for interface interaction in compressible multi-fluid flow

In this work, the HLLC Riemann solver, which is much more robust, simpler and faster than iterative Riemann solvers, is extended to obtain interface conditions in sharp-interface methods for compressible multi-fluid flows. For interactions with general equations of state and material interfaces, a new generalized Roe average is proposed. For single-phase interactions, this new Roe average does not introduce artificial states and satisfies the U-property exactly. For interactions at material interfaces, the U-property is satisfied by introducing ghost states for the internal energy. A number of numerical tests suggest that the proposed Riemann solver is suitable for general equations of state and has an accuracy comparable to iterative Riemann solvers, while being significantly more robust and efficient.