SPH Molecules - a model of granular materials

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1 SPH Molecules - a model of granular materials Tatiana Capone DITS, Univeristy of Roma (la Sapienza) Roma, Italy Jules Kajtar School of Mathematical Sciences Monash University Vic. 3800, Australia Joe Monaghan School of Mathematical Sciences Monash University Vic. 3800, Australia joe.monaghan@sci.monash.edu.au Abstract An SPH model of granular material is described. The model replaces standard SPH particles by small molecules each of which moves as a rigid body. We simulate dry granular materials forming piles and show that they have an angle of repose similar to that in nature. We simulate an avalanche by placing the molecules on a steep hillside and show that the avalanche runs onto a horizontal plane and eventually comes to rest. Finally, we simulate dambreaks using a mixture of granular and liquid SPH particles. The effective viscosity of the mixture increases as the concentration of the granular materials increases. The increase is small for a concentration less than 0.5 but increases very rapidly for concentrations above this. These results are similar to those for a Bingham rheology. I. INTRODUCTION This paper is concerned with the simulation of problems involving granular material or powders combined with fluids. They include geophysical phenomena such as mud flows and submarine landslides, and industrial phenomena such as the flow of chocolate or bitumen. The combination of a granular material and a fluid results in a material with a non-linear rheology. The rheology may be time dependent when the granular material breaks up, or when it becomes heated, or when the random motion of the grains alters. In some cases the rheology can be approximated by that of Bingham, Close, and Herschel-Bulkley. The simulation problem then reduces to integrating the Navier-Stokes equations with a non linear viscous term. In the case of the Bingham and Herschel-Bulkley rheologies this has been carried out by Vola et al. (2004) using a Characteristic Galerkin method with a finite element decomposition, and in the case of SPH by Shalo and Lo (2003) and Hosseini et al. (2006). The situation with many geophysical materials is much less satisfactory because the rheology is not known. In this paper we consider a more general approach where the effects of the grains or powder is modelled by a much smaller number of SPH particles which behave like rigid bodies which we call molecules. The molecules consist of a small number of particles, typically 5. The interaction between each of the molecules, and between the molecules and the fluid must be modelled and we do this by using simple forces. This aspect of our method is similar to molecular dynamics or the discrete element method (DEM). The advantage of our method is that we can explore very complicated systems involving many different types of grains and fluids and, because we use SPH we can, in principle, find the equivalent continuum equations by taking the continuum limit of the SPH equations. II. THE TYPES OF MOLECULES Fig. 1. Two examples of the molecules we use. Each filled circle marks the site of a particle of the molecule. In Fig.1 we show two examples of the molecules we used for our two dimensional simulations. Each filled black circle in figure 1 indicates a particle of the molecule, and is the site of a force which acts on particles whether from another molecule, a liquid SPH particle or a boundary force particle. The black lines in figure 1 (called legs in the following) emphasize the geometry. A molecule moves as a rigid body according to the total force and torque exerted on the particles of the molecule. The force/mass f jk exerted by a particle k of a molecule on a particle j of another molecule, boundary particle or fluid particle, is radial and has the form Fig. 2. A pile of granular material modelled by rectangles. Note the corrugated base and the fact that some molecules have fallen into the corrugations. 138

2 f jk = B 2m k rjk 2 K(r jk /h jk )r jk, (1) m j + m k where r jk = r j r k (this notation for vectors is used throughout this paper), h jk = 1 2 (h j + h k ), m j is the mass of particle j, h for a particle of a molecule has a length equal to 1.5 times the leg length of the molecule, and K(q) is the one dimensional Wendland (1995) kernel defined by K(q) = 5 8 (1 + 3q)(1 q)3, (2) for q 1 and zero otherwise. The constant B is taken as gd where g is the gravitational acceleration and D is the initial depth of the material. For a molecule of mass M, composed of K particles, the mass m of each particle of the body is M/K. The moment of inertia is calculated from the positions of the particles of the body relative to the centre of mass. Fig. 3. A closeup of the pile of rectangular molecules. Note the gaps in the pile and the corrugated base. All boundaries are formed by boundary force particles which interact with the particles of the molecules according to the force defined by (1). The boundary particle j interacts with fluid particles a according to the force/mass f aj = B 2m j raj 2 H(r aj /h aj )r aj, (3) m j + m k where H(q) has the value H(q) = 5 64 (1 + 3q/2)(2 q)3, (4) for q 2 and is zero otherwise. H therefore has the same range as the kernel for the fluid particles. This function produces smooth forces normal to the surface with high accuracy. For example the errors (a) relative variation in the force for a constant distance from the boundary, and (b) ratio of tangential force to normal force, are < 10 4 when the spacing of the boundary force particles is 1/2 that of the liquid particles rd ERCOFTAC SPHERIC workshop on SPH applications Although we only consider straight line boundaries in this paper, we have shown that these boundary force particles give excellent results for fluids inside or outside curved surfaces. III. PILES OF GRANULAR MATERIAL The most fundamental problem for granular materials is the formation of a pile at the typical angle of repose of around 20 degrees. We find that this is achieved provided the ground is roughened by the introduction of a small amplitude sinusoidal variation in height given by y = b sin (2πx/(4 b ), (5) where b is the spacing of the boundary particles. Fig. 2 shows a pile formed by dropping a sequence of groups of rectangular molecules onto the horizontal boundary. These molecules have leg length b = 0.25dp where dp is the initial separation of the centres of mass of the molecules. The rectangles have sides of length b and 3b. Fig. 2 shows that the molecules fall into the corrugation and get stuck. The flow is then over the jagged edge of these molecules. Fig. 3 shows a close up of the rectangular molecules. Because of the shape of the rectangles voids appear as in the case of naturally occurring piles of granular material. The piles formed from rectangles have an angle of repose of around 22 degrees and the piles formed from triangular molecules have an angle of repose around 17 degrees. A comprehensive study of the geometry of these piles and their relation to the experiments of Lajeunesse et al. (2005) is planned. A. Avalanches Avalanches are triggered by instabilities of soil, rock or snow on steep slopes. Fig. 4 shows an initial and final stages of an avalanche formed from rectangles running down a hillside at 45 degrees to the horizontal then onto a horizontal region. In the example shown the granular material does not penetrate into the courrugations because the boundary forces are large compared to the weight of the granular material. Studies of the runout length compared to that found in nature and in experiments have started. These results show that the molecules can produce the same kinds of effects as spherical granules with frictional forces. IV. DAM BREAKS WITH GRANULAR MATTER AND WATER In this section we describe a number of dam breaks with water and an increasing fraction of granular matter. The bottom of the tank is corrugated and the water is simulated using standard SPH equations with the pressure given by ) P = 1 7 c2 ρ 0 ( ( ρ ρ 0 ) 7 1 where c 2 = 100gD, g is the gravitational acceleration, and D is the initial depth of the dam material. We use a two dimensional, normalized Wendland (1995 )kernel which has the form W (r) = 7 64π (1 + 2r/h)(2 r/h)4 (7) (6)

3 Fig. 5. The pure water dam break after step 500. Note the corrugated bottom and the regular array of liquid SPH particles. Fig. 4. The initial and final states of an avalanche of rectangular molecules. Note that the topography is corrugated. for r 2h and is zero otherwise. We find the results using this kernel are better than those using the cubic spline because the fluid particles are less likely to form strings. A discussion of this point will be presented elsewhere. The viscous term (Monaghan 2005) has the form Π ab = αv sig ρ ab v ab r ab r ab, (8) where in the calculations described here vsig 2 = 100gD where D is the initial depth of the fluid in the dam break and g is the gravitational acceleration. The dam break is from a region with height 1m and width 1m and the tank is 4m long. The bottom is corrugated as for the examples where the granular material forms a pile. The molecules are rectangles with width b and length b 3 where b = 0.25dp and dp is the liquid particle spacing. The constant α is 0.03, and the effective kinematic viscosity with this Wendland kernel is αv sig h/8 and h = 1.5dp. We use 2500 particles in a 50x50 array, but each molecule comprises 5 particles. This means that the calculation time increases as the fraction of molecules increases. Fig. 5 shows the pure water dam break at step 500. The particles are in a highly ordered configuration but slightly disturbed at the base because of the corrugation. This dam break follows the usual pattern of running to the end of the tank, and forming a backwards breaking wave. Similar results are found for dam breaks where the fraction of molecules is less than 0.5. Fig. 6 shows a close up of the rectangular molecules and the liquid particles in part of the current when the fraction of granules is 0.25 but the total mass is the same as in the previous case. The molecules are well mixed with the fluid, except near the front of the current shown in Fig. Fig. 6. A close up of SPH particles for a dam break where the fraction of granular particles is The flow continues to the end of the tank, but the wave formation is weaker than for a pure water dam break. Fig. 8 shows the front of the flow when the fraction of molecules is In this case the dam break only extends to 3/4 of the tank width. One interpretation of this result is that the effective viscosity of the mixture has been greatly increased. In Fig. 9 we show the front of the dam break when there is no water. The molecules have only reached 0.5m beyond the initial front. Because the front of this pure granular dam break has an angle of repose greater than that in figure 2 we can expect that it will be unstable to large enough perturbations. These results show that the introduction of the molecules increases the viscosity of the fluid, a result which is in agreement with the experiments of Komatina and Jovanovic (1997) who studied the rheological properties of mixtures of kaolin and water in different concentrations. In Fig. 10 we show the position of the front of the dam break against time for dam breaks where the fraction of water in the mixture is 1.0, 0.5, 0.25 and 0.0. When the fraction is 0.5 the results 140 3

4 Fig. 7. The arrangement of the particles at the front of the dam break when the fraction of granular material is There is a roughly uniform mixing of the particles except very close to the front. Fig. 9. The front of the dam break of pure granular material. The flow has stopped. triangles). The results show that the rectangular and triangular molecules give results in close agreement. Fig. 8. The arrangement of the particles at the front of the dam break when the fraction of granular material is Note how the types of particles are not well mixed and some groups of molecules form cages around single water particles. (shown by filled triangles) are close to those for pure water (shown by filled circles). When the water fraction is 0.25 the speed of the front is reduced to 0.5 the value for pure water. When the water fraction is zero, the flow stops at a distance of 0.4 m beyond the initial front. This rapid increase in effective wiscosity with increase in the fraction of molecules is similar to the exponential variation found by Komatina and Jovanovic (1998). In Fig. 11 we show the speeds of the front for watermolecule mixtures consisting of rectangular molecules (shown by open triangles) and triangular molecules (shown by filled Fig. 10. The position of the front of various dam breaks against time with a mixture of granular material, modeled with rectangular molecules. The results for the pure water dam break are shown by filled circles, the filled triangles are for the case where the fraction of water is 0.5, the open circles are for the fraction 0.25 and the open triangles are for the case of no water. V. CONCLUSION The results described here, though preliminary, show that SPH algorithms using a mixture of molecules and liquid give results in qualitative agreement with those observed in nature and in laboratory experiments. The method is easy to implement and opens up the possibility of dealing with coastal erosion as well as submarine landslides. In the latter 141 4

5 Fig. 11. The position of the front against time for three dam breaks. The filled circles show the results for pure water. The open triangles are for rectangular molecules and the filled triangles are for triangular molecules. case it would be easy to simulate flows involving a mixture of different types of granular matter by using a mixture of molecules. We are currently working on the relationship between the parameters of the molecules and the effective viscosity. REFERENCES [1] D. Vola, F. Babik, J..C.latche, On a numerical strategy to compute gravity currents of non-newtonian fluids,j. Comp. Physics, 201, , (2004). [2] S. Shao, E. Y.H.Lo, Incompressible SPH mehod for simujlating Newtonian and non-newtonian flows with a free surface,adv. in Water Resources, 226, , (2003). [3] S. M.Hosseini, M. T.Manzari, S. K.hannani, A fully explicit three-step SPH algorithm for simulation of non-newtonian fluid flow, Int. J. of Num. Methods for Heat and Fluid Flow, 17(7), , (2006). [4] E. lajeunesse, J. B.Monnier, G. M.Homsy, Granular slumping on a horizontal surface,phys. of Fluids,17,103302, (2005). [5] D. Komatina, M. Jovanovic, Experimental study of steady and unsteady free surface flows with water-clay mixtures, J. Hydraulic Research, 35, , (1998). [6] J. J.Monaghan, emphsmoothed particle hydrodynamics, Reports on Progress in Physics, 68, , (2005). [7] H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. In Computat. Math. 4, , (1995)