Characteristic Piezocone Penetration Responses

Numerical Modeling of Suspension Flow Using the Lattice Boltzmann Method

M. Emin Kutay1 and Ahmet H. Aydilek2, Members, ASCE.

1Sr. Laboratory Manager, Turner-Fairbank Highway Research Center, Federal Highway Administration (FHWA), 6300 Georgetown Pike, Mclean, VA 22101; [email protected] 2Associate Professor, University of Maryland-College Park, Department of Civil and Environmental Engineering, College Park, MD, 20742, USA, [email protected]

ABSTRACT: The lattice Boltzmann (LB) method has been increasingly used in various civil engineering applications such as modeling fluid flow in granular materials and asphalt pavements. LB method is a numerical method for simulating viscous flow, where the Boltzmann equation of molecular dynamics is approximated and solved through discretization of physical space. Traditionally, solid particles in the fluid are assumed to be stationary during the fluid flow simulations in porous media. In this study, movements of solid inclusions in the fluid (i.e., suspensions) were modeled by implementing specific boundary conditions at solid-fluid interfaces. Details of these boundary conditions are described. Successful validation of model is presented for a moderately high Reynolds numbers.

INTRODUCTION

Significant progress has been made in the area of computational fluid dynamics in recent years and one of the reliable methods, the lattice Boltzmann (LB) approach, has been increasingly used in various engineering applications to model the flow of mono and multiphase fluids (Rothman and Zaleski 1998, Succi 2001, Chen and Doolen 2001, Hazi 2003, Kutay et al. 2007). The LB method has several advantages, including ease of implementation of boundary conditions and computational efficiency through parallel computing. The method naturally accommodates some of the boundary conditions such as a pressure drop across the interface between two fluids and wetting effects at the fluid-solid interface (Martys et al. 2001). It has been proven to be very accurate in simulating isothermal, incompressible flow at low Reynolds numbers (Succi 2001); however, recent work has highlighted its additional potential for simulating flow at high Reynolds numbers (Eggels 1996, Lu et. al 2002) An LB algorithm was developed in this study to model movements of solid inclusions in fluid, i.e., suspension flow. Specific boundary conditions at solid-fluid interfaces were developed and successful validation of the model was presented for a moderately high Reynolds numbers.

LATTICE BOLTZMANN METHOD

Theory

The LB method is a numerical method for simulating viscous fluid flow where it approximates the continuous Boltzmann equation by discretizing physical space with lattice nodes and velocity space by a set of microscopic velocity vectors (Maier et al. 1997). The time and space averaged microscopic movement of particles are modeled using molecular populations called distribution function, which defines the density and velocity at each lattice node. Specific particle interaction rules are set so that the Navier-Stokes equations are satisfied. The time dependent movement of fluid particles at each lattice node satisfies the following particle propagation equation:

Fi (x  e i , t 1)  Fi (x, t)   i (1)

where Fi, ei and i are the particle distribution function, the microscopic velocity and the collision function at lattice node x, at time t, respectively. The subscript i represent the lattice directions around the node as shown in Fig. 1a. The collision function represents the collision of fluid molecules at each node and has the following form (Bhatnagar et al. 1954):

F (x, t)  Feq (x, t)    i i (2) i 

eq  where Fi (x, t) is the equilibrium distribution function and is the relaxation time which is related to the viscosity of the fluid (   (2 1)/6 , where  is the kinematic viscosity). Equilibrium distribution functions for different models were derived by He and Luo (1997). The function is given in the following form for the two-dimensional LB model with nine microscopic velocity vectors (D2Q9):

 e u(x, t) e u(x, t)2 u(x, t)u(x, t) F eq (x, t) w (x, t) 1 i i i  i    2  4  2  (3)  c s 2c s 2c s  where  and u are the density and the macroscopic velocity of the node, wi is the th weight factor for i direction and cs is the lattice speed of sound (=1/ 3 for D2Q9 lattice). Weight factors (wi) for D2Q9 LB model are: w9 =4/9 for rest particle, wi=1/9 (1 i  4 ) for particles streaming to the face-connected neighbors and wi=1/36 ( 5  i  8 ) for particles streaming to the edge-connected neighbors. The two macroscopic properties, density ( ) and velocity (u) of the nodes, are calculated using the following relations: 9 9 F (x, t) e (x, t)  F (x, t)  i i  i u(x, t)  i1 (4) i1 (x, t)

There are three basic recurrent steps during simulation of viscous flow in a typical LB

2 algorithm: (1) propagation and collision, (2) application of boundary conditions, and (3) calculation of density, velocity and the new equilibrium distribution function. More details on the implementation of the D2Q9 LB method can be found in Kutay et al. (2006).

S o l i d n o d e s

B o u n c e - b a c k

F l u i d n o d e s

e e 2 e 6 5 e e e 3 9 1

e e e 8 7 4

FIG. 1. (a) Microscopic velocity directions for the D2Q9 lattice Boltzmann model. (b) Illustration of the solid boundary condition; bounce-back.

Boundary Conditions for Modeling Suspensions

In the particle propagation step of the algorithm, all components of the non- equilibrium distribution function are computed at each node except at nodes that are neighbors with solid nodes. At those neighboring nodes, the components of the distribution function ( Fi (x, t) ) expected to be migrating from the solid node are unknown. In general, unknown components of the distribution function at these nodes are calculated by the application of a no-slip boundary condition (Maier et al. 1996). It is also referred to as the bounce-back scheme, in which the distribution function components heading towards the solid nodes scatter directly back to the node (Fig. 1b). This boundary condition ensures that the solid particles remain stationary. In case of modeling moving solids in a fluid (e.g., suspension flow), on the other hand, the boundary conditions are more complicated. Pioneering work in this area was conducted by Ladd (1994-1, 2) and later on improved by Behrend (1995). The

Relaxed Bounce Back at the Nodes (RBBN) method described by Behrend (1995) is adopted in the current study for modeling the suspension flow. In this method, the solid particles are assumed to be rigid and all nodes within the solid boundary move with a rigid body rotation and translation. The velocities at the solid boundary nodes seen in Fig. 2 are calculated using the following equation given by Behrend (1995):

u b  u c  ω (xb  x c ) (5)

where ub is the velocity of a boundary node, uc is the velocity of the particle, ω is the angular velocity of the particle, xb is the coordinate of a boundary node, and xc is the coordinate of the center of mass of the particle. Before the initialization of the algorithm, the center of mass (xc) and moment of inertia (I) of each individual particle are computed. Following the propagation and collision step, total force and torque acting on each particle is computed at each time step of the algorithm. This total force exerted by the fluid on each solid boundary node is computed as follows (Mei et al. 2002):

F l u i d S o l i d

S o l i d b o u n d a r y n o d e s

FIG. 2. Illustration of solid boundary nodes

9 f (x , t)  e [F (x , t)  F (x , t)] b b  i i b iˆ b (6) i1

where fb (x b , t) , Fi (xb , t) , and ei are the force, the particle distribution function, and the microscopic velocity at the boundary node xb at time t, respectively (Fig 1a). The subscript iˆ on the right-hand-side of Eq. 6 represents the direction opposite to subscript i. For example, for i = 1, iˆ = 3 in Fig. 1a. After computation of forces at the boundary nodes of a particle, total force (fc) and torque (  ) acting on a particle are computed as follows:

fc  fb   (xb  xc )  fb (7) b1 b1

where N is the number of boundary nodes. Then, uc and ω of each particle for the next time step (t+1) are computed using the following formulation:

u c (t 1)  u c (t)  fc / m c ω(t 1)  ω(t)   / I (8)

where mc is the total mass of each particle and I is the moment of inertia of the particle.

Using these particle velocities (uc ) and angular velocities ( ω ), the velocities of the boundary nodes (ub ) for the next time step (t+1) are computed using Eq. 5. The locations of all pixels within the solid particle are updated using a rigid body translation and rotation. The rotation matrix (R) is first computed using the angular velocity (ω ) of each particle as follows:

cos(ω) -sin (ω) R    (9) sin (ω) cos(ω) 

Following this computation, the coordinates of all pixels within the boundaries of the particle are updated using R in the following formulation:

x p (t 1)  x c (t)  u c (t 1)  R (x p (t)  x c (t)) (10)

VALIDATION OF THE LB MODEL FOR SUSPENSION FLOW

The LB model presented herein was validated using the benchmark problem of flow past through a circular cylinder. Extensive literature reported on this problem has been frequently used for validating numerical models (Kawaguti and Jain 1966, Hamielec and Raal 1969, Dennis et al. 1970, Fornberg 1980, Ahmad 1996, Ding et al. 2004). In general, a flow-field is generated around a stationary cylinder for validation of numerical models (Fig. 3a). However, the current study deals with moving solids and the boundary conditions, and the modeling was accomplished by moving the circular solid in a stationary fluid (Fig. 3b) by incorporating the boundary conditions described above into the LB model.

d l e i f

w o l f

i u l x p p a stationary moving solid solid u x

stationary fluid

(a) (b)

FIG. 3. Examples of simulation methods for flow past circular cylinder: (a) stationary solid in a moving fluid and (b) moving solid in a stationary solid.

A cylinder with a diameter of D=2.0 m was placed in a 10-m high and 28.5 m-wide domain. The size of the domain was selected as 100 vertical lattice points and 285 horizontal lattice points, which corresponded to a space resolution (x=y) of 0.1 m/lattice. A horizontal velocity (ux) of 2 m/s was assigned to the solid. Simulations were run for two different fluids with viscosities of 0.1 Pa-s and 0.05 Pa-s. The 3 densities () of both fluids were 1 kg/ m . The moving cylinder velocity (ux=2 m/s) was kept constant that led to Reynolds numbers (Re=uxD/) of 20 and 40 for the two fluids. The time step (t) was 1/30 sec/lattice-time which corresponded to a relaxation time (  ) of 1 during the computations.

F l o w s t r e a m l i n e s , R e = 2 0 7 0 (a) 6 5 L=1.7 m 6 0 )

m 5 5 m 1 .

( 4 5 Y

3 0 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 X ( x 0 . 1 m m )

(b) L=4.4 m

F l o w s t r e a m l i n e s , R e = 4 0 7 0

) 5 5 m 1 .

0 5 0 x

3 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 X ( x 0 . 1 m ) FIG. 4. Streamlines computed after fluid flow simulations (a) Re=20 and (b) Re=40.

Fig. 4 shows the streamlines computed after the LB simulations. Two symmetric vortices are generated behind the cylinder as expected from the use of high Reynolds numbers (Panton 1996). The dimensionless reattachment lengths are compared with the L/r ratios reported in the literature in Table 1. Herein, L/r is the ratio of the distance of L measured from the downstream side of the cylinder to the point where the x-velocity changes its sign from positive to negative to the radius of the cylinder (r). A very good comparison is evident between the two sets of data.

Table 1. Comparison between the LB simulations and measurements.

Method L/r for Re=20 L/r for Re=40 LB Simulation (the current study) 1.7 4.4 Measurements reported by Braza et. 1.8 4.2 al (1986)

CONCLUSIONS

A lattice Boltzmann (LB) algorithm was developed to simulate suspension flows at high Reynolds numbers. At each time step of the algorithm, the normal and shear forces due to fluid flow are calculated at the perimeter of each solid particle, and the total force and torque were computed leading to a rigid body motion for each particle. The velocities of the nodes at the perimeter of the particle were fed into the fluid motion at the pore-solid interface. The accuracy of LB model was verified against measurements of flow past through a circular cylinder. An excellent agreement was observed between this solution and the LB simulations. The efforts presented herein show the applicability of the LB method in simulating suspension flow through stationary liquids. This preliminary model can be improved to develop algorithms in simulating flow in various geotechnical and pavement applications, including colloidal transport in high water content soils, clogging of porous asphalt and concrete pavements by fine soil, filtration of dredged sediments and slurries using granular filters.

REFERENCES

Ahmad, R.A. (1996). “Steady-state numerical solution of the Navier-Stokes and energy equations around a horizontal cylinder at moderate Reynolds numbers from 100 to 500.” Heat Transfer Engineering, 17: 31-81. Behrend, Q. (1995) “Solid–fluid boundaries in particle suspension simulation via the lattice Boltzmann method.” Phys Rev E, 52:1164–75. Bhatnagar, P., Gross, E. and Krook, M. (1954) “A model for collision process for gases. I. Small amplitude processes in charged and neutral one-component systems”, Phys. Rev. 94, 511. Braza, M., Chassaing, P., and Minh, H.H. (1986). “Numerical Study and Physical Analysis of the Pressure and Velocity Fields in the Near Wake of a Circular Cylinder.” J. Fluid Mech., 165:79-130. Chen S. and Doolen G.D. (2001). “Lattice Boltzmann method for fluid flows.” Annual Rev Fluid Mech. 30:329–64. Dennis, S. C. R. and Chang, G.Z. (1970). “Numerical solutions for steady flow past a circular cylinder at Reynolds number up to 100.” J. Fluid Mech., 42: 471-489. Ding, H., Shu, C., Yeo, K. S. and Xu, D. (2004). “Simulations of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least

square-based finite difference scheme.” Comput. Methods Appl. Mech. Engng., 193: 727-744. Eggels, J.M.G. (1996). “Direct and large-eddy simulation of turbulent fluid flow using the lattice-Boltzmann scheme.” Int J Heat Fluid Flow, 17:307–23. Fornberg, B. (1980). “A numerical study of steady viscous flow past a circular cylinder.” J. Fluid Mech., 98: 819-855. Hamielec, A. E. and Raal, J. D. (1969). “Numerical studies of viscous flow around circular cylinders.” Phys. Fluids, 12: 11-17. Hazi G. (2003). “Accuracy of lattice Boltzmann based on analytical solutions.” Phys Rev E , 67:056705-1–5-5. He X and Luo L. (1997). “Theory of lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann.” Phys Rev E 56:6811–7. Kawaguti, M. and Jain, P. (1966). “Numerical study of a viscous flow past a circular cylinder.” J. Phys. Society of Japan, 21: 2055-2062. Kutay, M.E., Aydilek, A.H., and Masad, E. (2006). “Laboratory validation of lattice Boltzmann method for modeling pore-scale flow in granular materials.” Computers and Geotechnics, 33: 381-395. Kutay, M.E., Aydilek, A.H., and Masad, E. (2007). “Computational and experimental evaluation of hydraulic conductivity anisotropy in hot-mix asphalt.” Int. J. of Pavement Engr., 8(1): 29-43. Ladd, A. J. C. (1994-1). “Numerical simulation of particular suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation.” J Fluid Mech , 271:285–309. Ladd, A. J. C. (1994-2). “Numerical simulation of particular suspensions via a discretized Boltzmann equation. Part 2. Numerical results.” J Fluid Mech, 271:311– 39. Lu ZY, Liao Y, Qian DY, McLaughlin JB, Derksen JJ, Kontomaris K. (2002). “Large eddy simulations of a stirred tank using the lattice Boltzmann method on a nonuniform grid.” J Comput Phys., 181:675–704. Maier R.S., Bernard R.S., and Grunau D.W. (1996). “Boundary conditions for the lattice Boltzmann method.” Phys. Fluids, 8(7):1788–801. Martys, N.S., Hagedorn, J.G., and Devaney, J.E. (2001). “Pore scale modeling of fluid transport using discrete Boltzmann method” Proceedings of Materials Science of Concrete, Special Volume: Ion and Mass Transport in Cement-Based Materials, Toronto, Canada, pp. 239–52. Panton, R.L. (1996). Incompressible flow, 2nd Ed., J. Wiley and Sons, Inc., New York. Rothman, D.H. and Zaleski, S. (1998). “Lattice-gas model of phase separation: interfaces, phase transitions, and multiphase flow.” Rev. Mod. Phys. 66(4):1471–9. Succi, S. (2001). The lattice Boltzmann equation: for fluid dynamics and beyond. Numerical mathematics and scientific computation series., Oxford-New York: Oxford University Press.