Mesoscale simulations of particulate flows with parallel distributed Lagrange multiplier technique Page: 10 of 47
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2.2 The numerical algorithm
The integration of governing equations is done using fractional time stepping approach at
time interval [t, t+1]. In the present algorithm, the velocity and pressure are cell-centered
quantities. The velocity is defined at integer multiplies of At, whereas the pressure is de-
fined at half-timesteps. The system of equations (1)-(3) with boundary conditions (3)-(5) are
solved numerically using operator-splitting technique that combines incompressibility condition,
advection-diffusion as a first step; the constraint of rigid-body motion in the particle domain
and the related distributed Lagrange multiplier technique as the next step.
First step: Solve Navier-Stokes equations in the entire domain. At the first step we solve fluid
equations of motion in the entire computational region and satisfy the provisional divergence-free
intermediate velocity field fn+1 using implicit pressure projection technique.
t - U" + -n+1 = Rn+1/2, (10)
V- = 0, (11)
where Rn+1/2 represents all terms on the right-sides of momentum equations except the pressure
gradient terms. An unsplit second-order Godunov procedure is used to approximate the nonlinear
advection term that appears in the momentum equations using both velocities defined at the
centers of the Cartesian grid as well as velocities defined at the cell faces. The MAC projection
method Bell et al. (1991) that corrects divergence of advection velocity along with MUSCL
advection scheme of Colella et al. (1985) are used to advect fluid. The Crank-Nickolson scheme
is used to compute diffusion terms. A divergence constraint is satisfied using an approximate
pressure projection method of Almgren et al. (1996). The density is computed as p = ps# +
pf (1 - 0), where 0 is the solid volume fraction which is equal to 1 in the solid domain and equal
to 0 in the fluid domain.
Second step: Rigid body projection in the particle domain. At the next stage, the constraint
of rigid body motion (in the form of Lagrange multiplier) is applied in the solid region. The
particle velocity us in a given cell is splitted in translational U and rotational Q parts as
us = U + Q x r, (12)
where r is a vector which connects a particle centroid and a center of the considered grid cell.
Particle velocities U and Q are computed by integration of the provisional velocity field u in
the solid domain as
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Kanarska, Y; Lomov, I & Antoun, T. Mesoscale simulations of particulate flows with parallel distributed Lagrange multiplier technique, article, September 10, 2010; Livermore, California. (digital.library.unt.edu/ark:/67531/metadc868095/m1/10/: accessed December 11, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.