Interdisciplinary Applied Mathematics

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Ц.3.2 The Force Coupling Method (FCM)


For a dilute limit with volume fraction much smaller than 10~3, particulate microflows can be modeled by simply adding a source term of the form


0


Sp = -J2 Fn6(x — Yn)


n


Re


FIGURE 14.16. Comparison of FCM against results from direct numerical simulation (SEM) for flow past an array of spheres. The particle Reynolds number Rep is plotted against the force Reynolds number, Ref, the latter being proportional to the drag coefficient. (Courtesy of G. Dent.) on the right-hand side of the Navier-Stokes equations, where the force on each particle is



















dVj(Yn) dt


and Yn is the vector coordinate associated with the particle’s center of mass; also, S(x — Yn) is the Dirac delta function. This is the so-called point-force model, which is not resolvable in numerical simulations of Navier-Stokes equations, and the results depend on the implementation details of the numerical scheme employed. It is used formally in Stokesian dynamics, but the    corresponding    equations    are    solved    analytically;    it    has    also    been


used in earlier gas-solid flows, e.g., dusty gases. The point-force approach does not directly take into account the finite particle size or interactions of particles in the flow. For example, this approach cannot capture the experimentally observed phenomena associated with acceleration of particles such as the DKT (drafting-kissing-tumbling) event. This phenomenon refers to nonlinear interactions between two spheres where the trailing sphere is first drawn into the wake of the leading sphere, it touches it, and then overtakes the leading sphere by tumbling around it (Fortes et al., 1987).

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