# Interdisciplinary Applied Mathematics

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To complete FCM, we need to update the locations of the virtual particles Yn from

dYj_

dt

Vi 7

and following an explicit (third-order Adams-Bashforth) time-stepping scheme we obtain

Y k+1 y k

At

—Vk — — Vk~l + —Vk~2 12 1 12 * 12 1

Here к refers to the time level, and At is the time step used. Note that in FCM we have Vk = uk.

FCM has been studied systematically by (Lomholt, 2000), who compared it with experimental results for one, two, and three particles. In Figure 14.17 we plot the results of Lomholt for a flow in a channel with two particles interacting,    with    gravity    the    body    force.    The    DTK    process    revealed    by

the experiments of (Fortes et al., 1987) was reproduced in experiments described in (Lomholt, 2000). We see that very good agreement for the particles, trajectory is obtained with the FCM predictions for this strongly nonlinear interaction.

In the    following,    we    compare    the    accuracy    of FCM    to    the    accuracy    of

direct numerical simulation (DNS) and to the distributed Lagrange multiplier (DLM) methods for a flow past a microsphere. All discretizations

FIGURE 14.17. Particle trajectories in (left) streamwise-normal directions and (right) streamwise-spanwise directions. Comparison of FCM predictions (lines) and experimental results (symbols). The geometry is a channel of width 10 mm with    the    walls    located    at    x2    = ±5 mm. The    particle Reynolds    number    is    1.71

for the leading particle (triangle) and 1.60 for the trailing particle (circle). All particles have the same radius of 1 mm. (Courtesy of S. Lomholt.)

are based on spectral elements. By DNS we mean that we enforce the noslip boundary condition on the particle directly as a Dirichlet condition. We consider the prototype problem of a flow past a sphere inside a threedimensional channel. The sphere is located near one wall of the channel, which has length 15, height 7, and width 7. The origin of the coordinate system is located    at    the center    of    the    channel,    so    that    the    flow    domain

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