20 Will-be-set-by-IN-TECH the mill main body diameter is 10 m while grid size is 75 mm. But with SPH, it is flexible to control the solver by assigning SPH particle probability of passing through, or by applying different sets of triangles to SPH and DEM particles. 6. Conclusions Three approaches to couple solid particle behavior with fluid dynamics have been described and three applications have been provided. For full coupling approaches DEM-CFD and DEM-SPH, they are physically equivalent, but may appear in different forms of equations. The governing equations have been carefully formulated. Numerical methods, difficulties and possible problems have been discussed in detail. The one-way coupling of CFD with DEM has been used in analysis of wear on lining structure and particle breaking probability during a pump operation. The DEM–CFD coupling has been applied to modeling fluidization bed. The multiphase DEM–SPH solver has been used in a wet grinding mill simulation. Each numerical approach has its strength and weakness with respect to modeling accuracy and computation cost. The final choice of best models should be made by application specialists on a case by case basis based on dominant features of physical phenomena and numerical models. 7. References Cundall, P. A. & Strack, O. D. L. (1979). Discrete numerical model for granular assemblies, Géotechnique 29: 47–64. Gao, D., Fan, R., Subramaniam, S., Fox, R. O. & Hoffman, D. (2006). Momentum transfer between polydisperse particles in dense granular flow, J. Fluids Engineering 128. Gao, D., Morley, N. B. & Dhir, V. (2003). Numerical simulation of wavy falling film flows using VOF method, J. Comput. Phys. 192(10): 624–642. Gera, D., Gautam, M., Tsuji, Y., Kawaguchi, T. & Tanaka, T. (1998). Computer simulation of bubbles in large-particle fluidized beds, Powder Technology 98: 38–47. Gera, D., Syamlal, M. & O’Brien, T. J. (2004). Hydrodynamics of particle segregation in fluidized beds, International Journal of Multiphase Flow 30: 419–428. Goldhirsch, I. (2003). Rapid granular flows, Annu. Rev. Fluid Mech. 35: 267–293. Goldschmidt, M. (2001). Hydrodynamic Modelling of Fluidised Bed Spray Granulation, Ph.D. Thesis, Twente University, Netherlands. Herbst, J. A. & Pate, W. T. (2001). Dynamic modeling and simulation of SAG/AG circuits with MinOOcad: Off-line and on-line applications, in D. Barratt, M. Allan & A. Mular (eds), Proceedings of International Autogenous and Semiautogenous Grinding Technology, Volume IV, Pacific Advertising Printing & Graphics, Canada, pp. 58–70. Herbst, J. A. & Potapov, A. V. (2004). Making a discrete grain breakage model practical for comminution equipment performance simulation, Powder Technology 143-144: 144–150. Hollow, J. & Herbst, J. (2006). Attempting to quantify improvements in SAG liner performance in a constantly changing ore environment, in M. Allan, K. Major, B. Flintoff, B. Klein & A. Mular (eds), Proceedings of International Autogenous and Semiautogenous Grinding Technology, Volume I, pp. 359–372. Huilin, L., Yurong, H. & Gidaspow, D. (2003). Hydrodynamics modelling of binary mixture in a gas bubbling fluidized bed using the kinetic theory of granular flow, Chemical Engineering Science 58: 1197–1205. 48 Hydrodynamics – Optimizing Methods and Tools Using DEM in Particulate Flow Simulations 21 Jenkins, J. T. & Savage, S. B. (1983). A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles, J. Fluid Mech. 130: 187–202. Landry, J. W., Grest, G. S., Silbert, L. E. & Plimpton, S. J. (2003). Confined granular packings: Structure, stress, and forces, Phys. Rev. E 67: 041303. Li, J. & Kuipers, J. A. M. (2002). Effect of pressure on gas-solid flow behavior in dense gas-fluidized study, Powder Tech. 127: 173–184. Monaghan, J. (1988). An introduction to SPH, Computer Physics Communications 48: 89–96. Monaghan, J. (1989). On the problem of penetration in particle methods, Journal of Computational Physics 82: 1–15. Monaghan, J. (1994). Simulating free surface flows with SPH, Journal of Computational Physics 110: 399–406. Monaghan, J. (1997). Implicit SPH drag and dusty gas dynamics, Journal of Computational Physics 138: 801–820. Monaghan, J. (2000). SPH without a tensile instability, Journal of Computational Physics 159: 290–311. Monaghan, J. & Kocharyan, A. (1995). SPH simulation of multi-phase flow, Computer Physics Communications 87: 225–235. Morris, J. P., Fox, P. J. & Zhu, Y. (1997). Modeling low reynolds number incompressible flows using SPH, Journal of Computational Physics 136: 214–226. Plimpton, S. (1995). Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117: 1–19. Potapov, A., Herbst, J., Song, M. & Pate, W. (2007). A dem-pbm fast breakage model for simulation of comminution process, in UNKNOWN (ed.), Proceedings of Discrete Element Methods, Brisbane, Australia. Qiu, X., Potapov, A., Song, M. & Nordell, L. (2001). Prediction of wear of mill lifters using discrete element methods, in D. Barratt, M. Allan & A. Mular (eds), Proceedings of International Autogenous and Semiautogenous Grinding Technology, Volume IV, Pacific Advertising Printing & Graphics, Canada, pp. 260–265. Rhie, C. & Chow, W. (1983). A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation, AIAA 21(11): 1525–1532. Rong, D. & Horio, M. (1999). DEM simulation of char combustion in a fluidized bed, in M. Schwarz, M. Davidson, A. Easton, P. Witt & M. Sawley (eds), Proceedings of Second International Conference on CFD in the Minerals and Process Industry, CSIRO Australia, CSIRO, Melbourne, Australia, pp. 65–70. Rusche, H. (2002). Computational fluid dynamics of dispersed two-phase flows at high phase fractions, Ph.D. Thesis, Imperial College London, UK. Savage, S. B. (1998). Analyses of slow high-concentration flows of granular materials, J. Fluid Mech. 377: 1–26. Silbert, L. E., Ertas, D., Grest, G. S. & et al. (2001). Granular flow down an inclined plane: Bagnold scaling and rheology, Phys. Rev. E 64: 051302. Srivastava, A. & Sundaresan, S. (2003). Analysis of a frictional-kinetic model for gas-particle flow, Powder Tech. 129: 72–85. Sun, J. & Battaglia, F. (2006a). Hydrodynamic modeling of particle rotation for segregation in bubbling gas-fluidized beds, Chemical Engineering Science 61: 1470–1479. Sun, J. & Battaglia, F. (2006b). Hydrodynamic modeling of particle rotation for segregation in bubbling gas-fluidized beds, Chemical Engineering Science 61(5): 1470–1479. URL: http://dx.doi.org/10.1016/j.ces.2005.09.003 49 Using DEM in Particulate Flow Simulations 22 Will-be-set-by-IN-TECH Syamlal, M. (1998). MFIX documentation: Numerical technique, Technical Note DOE/MC31346-5824, NTIS/DE98002029, National Energy Technology Laboratory, Department of Energy, Morgantown, West Virginia. See also URL http://www.mfix.org. Syamlal, M., Rogers, W. & O’Brien, T. (1993). MFIX documentation: Theory guide, Technical Note DOE/METC-95/1013, NTIS/DE95000031, National Energy Technology Laboratory, Department of Energy. See also URL http://www.mfix.org. Walton, O. R. (1992). Numeical simulation of inelastic, frictional particle–particle interaction, in M. C. Roco (ed.), Particulate Two-phase Flow, Butterworth-Heinemann, London, pp. 1249–1253. Walton, O. R. & Braun, R. L. (1986). Viscosity, granular-temperature, and stress calculations for shearing assemblies of inelastic, frictional disks, J. Rheol. 30: 949. Xiao, H. & Sun, J. (2011). Algorithms in a robust hybrid CFD-DEM solver for particle-laden flows, Communications in Computational Physics 9: 297–323. 50 Hydrodynamics – Optimizing Methods and Tools 3 Hydrodynamic Loads Computation Using the Smoothed Particle Methods Konstantin Afanasiev, Roman Makarchuk and Andrey Popov Kemerovo State University Russia 1. Introduction The study of wave fluid flows is now under special consideration in view of serious effects, caused by dams breaking and consequent formation of moving waves, their interaction with solids and structures, uprush on shore, etc. Thereby solving the problem of hydrodynamic loads estimation is important for designing the shape and stiffness of the structures, interacting with oncoming waves. Such problems, due to large deformations of free surfaces, are very complex, and meshless methods proved to be the most suitable for numerical simulation of them. Particle methods form the special class of meshless methods, which mainly based on the strong form of governing equations of gas dynamics and fluid dynamics. The peculiar representatives of particle methods are Smoothed Particle Hydrodynamics (SPH) (Lucy, 1977; Gingold & Monaghan, 1977) and Incompressible SPH (ISPH) (Cummins & Rudman, 1999; Shao & Lo, 2003; Lee et al., 2008). Large amount of papers, devoted to numerical simulations of free surface flows using SPH or ISPH, demonstrated a high degree of efficiency of both methods in obtaining the kinematic characteristics of flows, though it has been revealed, that ISPH shows a larger particle scattering at the stages, following the water impact, in comparison with the classic SPH, where particles are more ordered. However, dynamic characteristics of flows are still hard to compute, especially it concerns the classic SPH. The objective of the chapter is to analyze the capacity of the methods to compute pressure fields and hydrodynamic loads subsequently. 2. Governing equations The governing equations of fluid dynamics, including the Navier-Stokes equations and the continuity equation, in the case of the Newtonian viscous compressible fluids, are of the following form: 11 (); a aab ab p dv FT dt xx       (1) , a a dv dt x      (2) Hydrodynamics – Optimizing Methods and Tools 52 where ab 123 – numerical indices of coordinates, a v – components of the velocity vector, a F – components of the vector of volumetric forces density, ab  – Kronecker symbols, p and  – pressure and density of the fluid, correspondingly. Here the Einstein summation convention is assumed. The viscous stress tensor components are calculated by the formula (  - dynamic viscosity): 2 3 ab c ab ab ba c vv v T xx x           (3) For enclosing the system (1)-(3) one should make some assumptions about fluid properties. The original SPH method assumes the fluid to be weakly compressible, and therefore is applied to the system (1)-(3) with certain equation of state for enclosure. The most often used equation of state is the Theta form equation for barotropic processes (Monaghan et al., 1994): pB 0 1                 (4) Selecting the coefficient of volume expansion B one can obtain the effect of incompressible fluid. The ISPH method in contrast to the original SPH uses the model of incompressible fluid, what means ddt /0   . In that case the equation of state shouldn’t be considered and the enclosed system of governing equations takes the following form: aa a abb p dv v F dt xxx 1 ()         (5) a a v x 0     (6) 3. Smoothed particle methods 3.1 The basis of the methods The key idea of smoothed particle methods lies in discretization of the problem domain into a set of Lagrangian particles, which play the role of nodes in function approximation. For construction of approximation formulas in smoothed particle methods the exact integral representation with the Dirac  -function is used: ff d() ( )( )         rrrrr (7) The Dirac  -function is changed here by a compactly supported function W , called the kernel function, what allows to obtain the integral formula about the bounded domain: D ff Whd() ( ) ( )       rrrrr (8) The value h determines a size of support domain D of the function W and is called a smoothing length. Having a set of particles scattered about the problem domain  we Hydrodynamic Loads Computation Using the Smoothed Particle Methods 53 can estimate the value of the above integral with the quadrature (Lucy, 1977; Gingold & Monaghan, 1977): n j si j i j j j m ff Wh 1 () () ( )      rrrr (9) where n is a number of particles, determined as “nearest neighbours” of the i -th particle within the support domain D . Two particles i and j are called neighbouring or interacting particles, if the distance between their centers does not exceed kh , where k depends on the type of kernel function and ij hhh()/2 . jjj m   r - radius-vector, mass and density of the j -th particle, correspondingly. A simple formula for the gradient of a function has the form: 1 () () ( ) n j si j i j j j m ff Wh      rr rr (10) 3.2 Kernel function As kernel function is a keystone of smoothed particle methods a great attention is paid to construction of new types of kernels. Till now a large amount of different types of kernel functions have been developed. All of them should satisfy the following basic conditions: - Wh kh()0      rr rr - Whd()1      rr r - 0 lim ( ) ( ). h Wh      rr rr Here for the problems, simulated with SPH, the original Monaghan’s cubic spline is utilized (Monaghan et al., 1994):  qq q Wh q q h q 23 3 2 23 20 1 15 () 612 2 7 02                 rr (11) where q h    rr . As it was pointed out (G.R. Liu & M.B. Liu, 2008) the approximations of functions based on the kernels that haven’t smooth second derivative are too sensitive to particle scattering. It plays a crucial role for the ISPH method as elliptic Poisson equation is solved for obtaining a pressure field. That is why in numerical simulations using ISPH the fourth-order spline has been used (Morris, 1996; Lee et al., 2008):    qq q q qq 1/2q Wh h q 3/2q q 44 4 44 2 4 5/2 5 3/2 10 1/2 , 0 1/2 96 5/2 5 3/2 , 3/2 () 1199 5/2 , 5/2 0, 5/2                       rr (12) Hydrodynamics – Optimizing Methods and Tools 54 3.3 Approximation of governing equations For approximation of gradient terms in equations (1) or (5) the original formula (10) may be applied. However, it is usually implemented for derivation of new forms of gradient approximations. In numerical simulations the following form is commonly used: n j i ij iij j ij p p p mWh 22 1 1 (,)             rr (13) This formula has an advantage of being symmetric in relation to interacting particles and thus conserves total momentum of a system of particles, representing the problem domain. Besides it gives more stable results of numerical simulations in comparison to (10). For a divergence of a velocity field in the continuity equation (2) the following expression is usually applied: 1 1 ()( ) n i jiji ij j mWh i         vvvrr (14) The above form gives a zero-valued first derivatives for a constant field. Using (13) for approximation of gradient of a function one can obtain the following discrete representation for viscous term in equation (1): 22 1 1 () n j i jiij j ij i mWh                 T T Trr (15) Normal and tangent components of viscous stress tensor T i are defined by following expressions similar to (14) (G.R. Liu & M.B. Liu, 2008): 1 1 ()()()() 2 ()( ) 3 n j ab a a b b b a ijiiijjiiij j j n j ab jii ij j j m TvvWhvvWh m Wh                        rr rr vv rr (16) As it will be pointed out in section 3.4 the pressure Poisson equation need to be solved in the ISPH method. There are some ways to obtain the approximations of second derivatives in smoothed particle methods. One way consists in directly deriving the formula in a similar manner as for first derivative (10). The idea of the other is in subsequent implementation of a gradient formula (13) and a divergence of vector field (14). However these ways proved to be too sensitive to inhomogeneous particle distribution and result in non-physical oscillations of pressure field. So the approximation of the first derivative in terms of the SPH method and its finite difference analogue are usually applied together according to Brookshaw’s idea (Brookshaw, 1985). Based on it some different forms of Laplacian operator were derived (Cummins & Rudman, 1999; Shao & Lo, 2003; Lee et al., 2008). Here for numerical simulations the form of Lee (Lee et al., 2008) is used: n ijiji ij ij i j ij p pWh pm 2 1 () 2            rr rr rr (17) Hydrodynamic Loads Computation Using the Smoothed Particle Methods 55 The approximation formulas for viscous forces in ISPH are obtained in a similar way and may take different forms (Cleary & Monaghan, 1999; Shao & Lo, 2003). Here for numerical simulations the following viscous term by Morris (Morris et al., 1997) is utilized:  n ijijiij j i j ij j i ij Wh m 2 1 ()                  rr rr vvv rr (18) 3.4 Time integration In the original SPH method for time integration the "predictor-corrector" scheme is commonly used: "predictor": nn n ii i nn n ii i td dt td dt 12 1 12 1 (2)( ) (2)( )                    vv v (19) "corrector": nn n ii i nn n ii i td dt td dt 12 12 12 12 () ().                 vv v (20) The new radius-vectors of particles on (1)n  -th time step are calculated using the Euler integration scheme: nn n ii i tdt 112 ()     rr v (21) For time integration of motion equations in the ISPH method the split step scheme is applied (Yanenko, 1960; Chorin, 1968). According to its idea time integration process is splitted into convection-diffusion and pressure contribution. So the first step of the scheme for preliminary velocity values takes the from: n i t            vg v v (22) Projecting the preliminary velocity values onto a null-divergence field one can obtain: nn ii p t 11 1 ,      vv (23) provided the pressure field on n(1)  -th time step is calculated through the pressure Poisson equation (Lee et al., 2008): n i i p t 1 ,       v (24) Hydrodynamics – Optimizing Methods and Tools 56 where the velocity divergence at right hand side of above equation is calculated using formula (14). The radius-vectors of particles on n(1)  -th time step can be get out of the following formula according to Euler explicit integration scheme: nnn iii t 11  rrv (25) The equation (24) is reduced to the system of linear algebraic equations with symmetric matrix. For solving this system the preconditioned generalized minimum residual method (PGMRES) is applied (Saad, 2003). 3.5 Free surface For identification of particles on the free surface, one can apply some different ways. One of such ways is using the geometrical Dilts algorithm (Dilts, 2000), based on the fact, that each particle has its size, commonly determined by the smoothing length. The other way is detection of particles, satisfying the inequality (Lee et al., 2008):  1 (,)2 n j ij i ij j j m Wh        rr rr (26) as free surface particles have less nearest neighbors in comparison with the inner ones. Here the Dilts algorithm is used for the original SPH method and the formula (26) for ISPH. For free surface particles the Dirichlet condition is imposed: p 0  . For the original SPH it means that free surface particles has the zero pressure, not the pressure obtained out of the equation of state as for inner fluid particles. As the pressure Poisson equation (PPE) is solved in the ISPH method for obtaining pressure field, the Dirichlet condition is embedded into the matrix of the system of linear algebraic equations (SLAE), which is the discrete representation of PPE. This procedure conserves the symmetry of matrix of SLAE. 3.6 Solid boundary In smoothed particle methods the most commonly way of imposing conditions at solid boundaries is the virtual particle method, divided into two basic types. The first type – Monaghan virtual particles method (Monaghan et al., 1994). The virtual particles are located along the solid boundary in a single line, don’t change their characteristics in time, and effect on the fluid particles by means of a repulsive force, based on certain interaction potential. The most popular among researchers is the Lennard-Jones potential. The second type – Morris virtual particles (Morris et al., 1997). These particles are located along the solid boundary in several lines. The number of the lines depends on the smoothing length of particles of the fluid. This allows solving one of the main problems of the SPH method – asymmetry of the kernel function near the boundaries. The effect of the Morris particles on the fluid ones differs from the effect of Monaghan particles by the fact, that there is no need in using any interaction potential. Instead of this, values of the characteristics in the Morris particles are calculated on the basis of their values in particles of the fluid. Here for imposing solid boundary conditions on velocity the Morris virtual particles are used for both methods. In ISPH the Morris virtual particles are also implemented for imposition of Neumann boundary conditions on solid walls, that is /0pn   (Koshizuka et al., 1998; Lee et al., 2008). The procedure of embedding these conditions into the matrix of SLAE breaks its symmetry. Therefore, as it was mentioned in section 3.4, the PGMRES solver is utilized. [...]... boundaries Г 1 and Г 3 (fig 3) On horizontal solid walls Г 2 and Г 4 the slip condition is set (the zero-valued velocity vector) Fig 3 Problem domain for Poiseuille flow Within the problem domain  the fluid motion is described with the simplified momentum equation: 60 Hydrodynamics – Optimizing Methods and Tools dv dt  Pout  Pin L   d2 v (34 )  dy 2 where Pin , Pout - the pressure at Г 1 and Г 3 accordingly;... p(0) is initial pressure and p(T ) is the pressure at any other moment T (31 ) 58 Hydrodynamics – Optimizing Methods and Tools In the numerical computations the value of the integral (31 ) is estimated by the formula: Ps   p j (T )  p j (0) (32 ) jPB where Pb is a set of solid boundary particles 5 Nearest neighbour search In numerical simulations using the smoothed particle methods it is necessary... for every particle j its interacting particles, as all physical characteristics of the fluid are estimated over the values at neighbouring particles according to the formula (9) For each fluid particle j its smoothing length h j is set, determining the radius of interaction with neighbours As it is clear from section 3. 1 in smoothed particle methods if particle i interacts with particle j then particle... 1995): y sin  v( y )  g( H  )y 2  (39 ) In simulations by the smoothed particle methods the non-stationary equations were used and the convergence of the non-stationary solutions obtained by SPH and ISPH methods to the stationary analytic solution (39 ) are considered Parameters used in numerical simulations: L  H  10 3 m , density of the fluid   1000 kg/m3 , kinematic viscosity   10 6 m2 /s... RungeKutta method Table 3 shows the comparison of numerical errors by the original SPH method and by ISPH for t  0.8 s with different numbers of fluid particles, which corresponds to the moment of time for the relation of semi-axes of ellipse 1:2 a) b) c) d) Fig 7 Droplet problem: a) t  0 43 s , b) t  08 s , c) t  1.16 s , t  1.51 s 64 Hydrodynamics – Optimizing Methods and Tools N Numerical error... Industrial and Applied Mathematics, 10.09.2011, Available from http://www-users.cs.umn.edu/~saad/books.html 68 Hydrodynamics – Optimizing Methods and Tools Shao, S & Lo, E.Y.M (20 03) Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface Advances in Water Resources, Vol 26, pp 787–800 Slezkin, N.A (1995) The dynamics of a viscous incompressible fluid, Technical and. .. the particle approximation is marked by the index a The subscript will be avoided from now on The presence of these two variables allows SPH to be easily and conveniently applied to hydrodynamics problems The smooth estimate eq (7) can be referred to a generic particle occupying the position xi , as follow  f  x i    fi   N mk fk Wik k  1 k  (8) 72 Hydrodynamics – Optimizing Methods and Tools. .. along the X -axis and finite height H along the Y -axis a) b) Fig 5 Problem domain: a) initial, b) simplified The infinity of the channel is modeled by the algorithm described for Poiseuille flow in section 6.1 Provided Fx  g sin  , equation of motion with slip boundary conditions is written as: 0  2v  g sin  y 2 (37 ) 62 Hydrodynamics – Optimizing Methods and Tools v|Г 1  0 (38 ) As was mentioned... compressible and truly incompressible algorithms for the SPH mesh free particle method Journal of Computational Physics, Vol 227, pp 8417–8 436 Leonardo, Di G.S.; Jaime, K.; Eloy, S.; Yasmin, M & Anwar, H (20 03) SPH simulations of time-dependent Poiseuille flow at low Reynolds numbers Journal of computational physics, Vol 191, No 2, pp 622- 638 Liu, G.R & Liu, M.B (20 03) Smoothed particle hydrodynamics: ... (36 ) and the points show the results by SPH for 2500 fluid particles Approximately at t  0.6 s (fig 4 b) flow within the channel becomes stationary In table 1 the numerical errors by SPH and ISPH are compared a) Fig 4 Velocity profile for Poiseuille flow: a) t  0.02 s , b) t  0.6 s b) 61 Hydrodynamic Loads Computation Using the Smoothed Particle Methods N 225 625 900 1600 2500 36 00 12.9 6.9 5.0 3. 3 . Physics 9: 297 32 3. 50 Hydrodynamics – Optimizing Methods and Tools 3 Hydrodynamic Loads Computation Using the Smoothed Particle Methods Konstantin Afanasiev, Roman Makarchuk and Andrey Popov. 1/2q Wh h q 3/ 2q q 44 4 44 2 4 5/2 5 3/ 2 10 1/2 , 0 1/2 96 5/2 5 3/ 2 , 3/ 2 () 1199 5/2 , 5/2 0, 5/2                       rr (12) Hydrodynamics – Optimizing Methods and. momentum equation: Hydrodynamics – Optimizing Methods and Tools 60 out in PP dv d v dt L d y 2 2      (34 ) where in out P P, - the pressure at Г 1 and Г 3 accordingly; , L,  