A turbulence model for smoothed particle hydrodynamics

The aim of this paper is to devise a turbulence model for the particle method Smoothed Particle Hydrodynamics (SPH) which makes few assumptions, conserves linear and angular momentum, satisfies a discrete version of Kelvin's circulation theorem, and is computationally efficient. These aims are achieved. Furthermore, the results from the model are in good agreement with the experimental and computational results of Clercx and Heijst for two dimensional turbulence inside a box with no-slip walls. The model is based on a Lagrangian similar to that used for the Lagrangian averaged Navier Stokes (LANS) turbulence model, but with a different smoothed velocity. The smoothed velocity preserves the shape of the spectrum of the unsmoothed velocity, but reduces the magnitude for short length scales by an amount which depends on a parameter $\epsilon$. We call this the SPH-$\epsilon$ model. The effectiveness of the model is indicated by the fact that the second order velocity correlation function calculated using the smoothed velocity and a coarse resolution, is in good agreement with a calculation using a resolution which is finer by a factor 2, and therefore requires 8 times as much work to integrate to the same time.Comment: 34 pages, 11 figure

Smoothed Particle Magnetohydrodynamics (some shocking results...)

There have been some issues in the past in attempts to simulate magnetic fields using the Smoothed Particle Hydrodynamics (SPH) method. SPH is well suited to star formation problems because of its Lagrangian nature. We present new, stable and conservative methods for magnetohydrodynamics (MHD) in SPH and present numerical tests on both waves and shocks in one dimension to show that it gives robust and accurate results.Comment: Kluwer latex, 6 pages, 3 figures; Proceedings of the International Workshop "Magnetic Fields and Star Formation: Theory vs Observations", Madrid, 21-25 April 2003. Revised version accepted to proceedings (exact solutions added, other minor changes

Smoothed Particle Magnetohydrodynamics II. Variational principles and variable smoothing length terms

In this paper we show how a Lagrangian variational principle can be used to derive the SPMHD (smoothed particle magnetohydrodynamics) equations for ideal MHD. We also consider the effect of a variable smoothing length in the SPH kernels after which we demonstrate by numerical tests that the consistent treatment of terms relating to the gradient of the smoothing length in the SPMHD equations significantly improves the accuracy of the algorithm. Our results complement those obtained in a companion paper (Price and Monaghan 2003a, paper I) for non ideal MHD where artificial dissipative terms were included to handle shocks.Comment: 14 pages, 4 figures, accepted to MNRA

SPH simulations of turbulence in fixed and rotating boxes in two dimensions with no-slip boundaries

In this paper we study decaying turbulence in fixed and rotating boxes in two dimen- sions using the particle method SPH. The boundaries are specified by boundary force particles, and the turbulence is initiated by a set of gaussian vortices. In the case of fixed boxes we recover the results of Clercx and his colleagues obtained using both a high accuracy spectral method and experiments. Our results for fixed boxes are also in close agreement with those of Monaghan1 and Robinson and Monaghan2 obtained using SPH. A feature of decaying turbulence in no-slip, square, fixed boundaries is that the angular momentum of the fluid varies with time because of the reaction on the fluid of the viscous stresses on the boundary. We find that when the box is allowed to rotate freely, so that the total angular momentum of box and fluid is constant, the change in the angular momentum of the fluid is a factor ~ 500 smaller than is the case for the fixed box, and the final vorticity distribution is different. We also simulate the behaviour of the turbulence when the box is forced to rotate with small and large Rossby number, and the turbulence is initiated by gaussian vortices as before. If the rotation of the box is maintained after the turbulence is initiated we find that in the rotating frame the decay of kinetic energy, enstrophy and the vortex structure is insensitive to the angular velocity of the box. On the other hand, If the box is allowed to rotate freely after the turbulence is initiated, the evolved vortex structure is completely different

Lateralised sleep spindles relate to false memory generation

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Sleep is known to enhance false memories: After presenting participants with lists of semantically related words, sleeping before recalling these words results in a greater acceptance of unseen “lure” words related in theme to previously seen words. Furthermore, the right hemisphere (RH) seems to be more prone to false memories than the left hemisphere (LH). In the current study, we investigated the sleep architecture associated with these false memory and lateralisation effects in a nap study. Participants viewed lists of related words, then stayed awake or slept for approximately 90 min, and were then tested for recognition of previously seen-old, unseen-new, or unseen-lure words presented either to the LH or RH. Sleep increased acceptance of unseen-lure words as previously seen compared to the wake group, particularly for RH presentations of word lists. RH lateralised stage 2 sleep spindle density relative to the LH correlated with this increase in false memories, suggesting that RH sleep spindles enhanced false memories in the RH

Direct Numerical Simulation of decaying two-dimensional turbulence in a no-slip square box using Smoothed Particle Hydrodynamics

This paper explores the application of SPH to a Direct Numerical Simulation (DNS) of decaying turbulence in a two-dimensional no-slip wall-bounded domain. In this bounded domain, the inverse energy cascade, and a net torque exerted by the boundary, result in a spontaneous spin up of the fluid, leading to a typical end state of a large monopole vortex that fills the domain. The SPH simulations were compared against published results using a high accuracy pseudo-spectral code. Ensemble averages of the kinetic energy, enstrophy and average vortex wavenumber compared well against the pseudo-spectral results, as did the evolution of the total angular momentum of the fluid. However, while the pseudo-spectral results emphasised the importance of the no-slip boundaries as generators of long lived coherent vortices in the flow, no such generation was seen in the SPH results. Vorticity filaments produced at the boundary were always dissipated by the flow shortly after separating from the boundary layer. The kinetic energy spectrum of the SPH results was calculated using a SPH Fourier transform that operates directly on the disordered particles. The ensemble kinetic energy spectrum showed the expected k-3 scaling over most of the inertial range. However, the spectrum flattened at smaller length scales (initially less than 7.5 particle spacings and growing in size over time), indicating an excess of small-scale kinetic energy

Evaluations of topological Tutte polynomials

We find new properties of the topological transition polynomial of embedded graphs, $Q(G)$. We use these properties to explain the striking similarities between certain evaluations of Bollob\'as and Riordan's ribbon graph polynomial, $R(G)$, and the topological Penrose polynomial, $P(G)$. The general framework provided by $Q(G)$ also leads to several other combinatorial interpretations these polynomials. In particular, we express $P(G)$, $R(G)$, and the Tutte polynomial, $T(G)$, as sums of chromatic polynomials of graphs derived from $G$; show that these polynomials count $k$-valuations of medial graphs; show that $R(G)$ counts edge 3-colourings; and reformulate the Four Colour Theorem in terms of $R(G)$. We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of $P(G)$ and $R(G)$.Comment: V2: major revision, several new results, and improved expositio

Particle Methods: Past, Present and Future

Particle methods were first developed to handle the advection term in the equations of motion of fluids. They can do this easily because the particle trajectories are the characteristics of the hyperbolic part of the equations (the derivative following the motion), and the particles carry the fluid properties as they move. The equations of motion have other terms involving the spatial derivatives and these were originally calculated using a grid as in the Particle in Cell method (PIC) due to Harlow. A related method due to Eastwood is the EPICmethod where virtual particles are found each time step with the property that they will arrive exactly on a node or vertex at the end of the step. This method called the quasi-lagrangian method in atmospheric sciences. A different method, and the one I shall talk about, is Smoothed Particle Hydrodynamics or SPH. In this method the spatial derivatives are calculated directly from the particles using a simple interpolation. The resulting equations of motion look like the equations of molecular dynamics and SPH can therefore be considered as a valid way of approximating the continuum equations or as a model of the underlying molecular dynamics. The advantage of working directly with the particles is that when splash or fragmentation occurs it is easy to follow the transition from the continuum to the fragmented state. The original SPH calculations were applied to astrophysics, especially star formation, where there are huge changes in density, and a great deal of the space in which the stars move is empty of matter. Because most of the action is where the particles are it is efficient to work directly with the particles and this makes SPH codes very efficient. SPH can be generalized easily to adjust the resolution as the particles move together in a region of increasing density. No other method has this simplicity. Further improvements were associated with deriving the equations of motion from a Lagrangian which led to improved conservation properties. To extend the method to nearly incompressible fluids such as water, the only change required is to use an equation of state which is sufficiently stiff to make the density fluctuations negligible. In addition rigid boundaries must be included and this has been done by either using boundary force particles, or by using ghost particles. Free surfaces do not need a special treatment. With these techniques it has been possible to simulate waves breaking on beaches, dam breaks and a wide range of problems involving rigid bodies moving in one or more fluids and, more exotically, special effects in movies. In my presentation I will describe many of these applications. In particular I will describe new work on the swimming of linked rigid bodies which opens up the detailed application of SPH to swimming robots and swimming fish, even those that leap out of the water. The interested reader will find an extensive discussion of the theory in a recent review (Monaghan, Reports on Progress in Physics ,2005)