A truncation error analysis has been developed for the approximation of spatial derivatives in Smoothed Particle Hydrodynamics (SPH) and related first-order consistent methods such as the first-order form of the Reproducing Kernel Particle Method. Error is shown to depend on both the smoothing length h and the ratio of particle spacing to smoothing length, ∆x/h. For uniformly spaced particles in one dimension, analysisshows that as h is reduced while maintaining constant ∆x/h, error decays as h 2 until a limiting discretisation error is reached, which is independent of h. If ∆x/h is reduced while maintaining constant h (i.e. if the number of neighbours per particle is increased), error decreases at a rate * nathan.quinlan@nuigalway.ie in SPH, and shows that the roles of both h and ∆x/h must be considered when choosing particle distributions and smoothing lengths.

SUMMARYNumerous methods are available for the modelling of viscous stress terms in smoothed particle hydrodynamics (SPH). In this work, the existing methods are investigated systematically and evaluated for a range of Reynolds numbers using Poiseuille channel and lid-driven cavity test cases. The best results are obtained using two methods based on combinations of finite difference and SPH approximations, due to Morris et al. and Cleary. Gradients of high-valued functions are shown to be inaccurately estimated with standard SPH. A method that reduces the value of functions (in particular, pressure) before calculating the gradients reduces this inaccuracy and is shown to improve performance. A mode of instability in Poiseuille channel flows, also reported in other works, is examined and a qualitative explanation is proposed. The choice of boundary implementation is shown to have a significant effect on transient velocity profiles in start-up of the flow. The use of at least linear extrapolation for in-wall velocities is shown to be preferable to mirroring of velocities. Consistency corrections to the kernel are also found to result in significant accuracy and stability improvements with most methods, though not in all cases.

Inflow and outflow boundary conditions are essential for the application of computational fluid dynamics to many engineering scenarios. In this paper we present a new boundary condition implementation which enables the simulation of flow through permeable boundaries in the Lagrangian mesh-free method Smoothed Particle Hydrodynamics (SPH). Each permeable boundary is associated with an inflow or outflow zone outside the domain, in which particles are created or removed as required.The analytic boundary condition is applied by prescribing the appropriate variables for particles in an inflow or outflow zone, and extrapolating other variables from within the domain. Characteristic-based nonreflecting boundary conditions, described in the literature for mesh-based methods, can be implemented within this framework. Results are presented for simple one-dimensional flows, quasi-one-dimensional compressible nozzle flow, and two-dimensional flow around a cylinder at Reynolds numbers of 40 and 100, and Mach number of 0.1. These results establish the capability of SPH to model flows through open domains, opening a broad new class of applications.

SUMMARYA framework for adaptively inserting and removing particles with smoothed particle hydrodynamics (SPH) has been developed. A number of SPH variants were examined for use in an adaptive method. A minimum of linear consistency in the method has proven critical. Algorithms for particle placement and reassignment are discussed and results for a shock tube problem are shown.

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