Numerical investigation of the history and added mass forces acting on a spherical micro-particle in rectilinear motion in the case of finite Reynolds numbers

A.N. Nuriev ^a* , O.N. Zaitseva ^a**, A.I. Yunusova ^b***

^a Kazan Federal University, Kazan, 420008 Russia
^b Kazan National Research Technological University, Kazan, 420015 Russia

E-mail: *nuriev an@mail.ru, **olga fdpi@mail.ru, ***yunusova24@rambler.ru

Received September 18, 2017

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Abstract

In this paper, the study of components (quasi-stationary, history, and added mass forces) of the hydrodynamic force in several cases of rectilinear motion of the sphere at relatively low Reynolds numbers (5 < Re < 300) has been carried out. Stationary motion, linearly accelerated motion, uniform motion of the sphere following the step change of velocity has been considered. The calculation of the forces acting on the sphere has been performed based on the numerical solution of the flow problem. The fluid motion has been described by the complete non-stationary system of the Navier–Stokes equations. The methods for extraction of various components of the hydrodynamic force have been investigated, as well as the applicability of simplified models describing their behavior. The estimates of the history force contribution to the total force following the step change of motion velocity have been presented. It has been shown that the law of decay of this component is different for the cases of unidirectional and reverse motion due to different transition flows. A universal approach to determine the force of the added mass for the case of large accelerations has been proposed. On its basis, the analysis of the accelerated motion for the cases of uniform and piecewise linear acceleration laws has been carried out. The results of the analysis confirm the hypothesis of the linear nature of the added mass force acting on the sphere.

Keywords: viscous fluid, rectilinear motion of sphere, quasi-stationary force, history force, added mass force, numerical simulation

Acknowledgments. The study was supported by the Russian Foundation for Basic Research (project no. 16-31-00462-mol_a).

Figure Captions

Fig. 1. The computational grid near the sphere in xOy plane.
Fig. 2. Changes of the resistance forces depending on time following the step increase in the velocity for Re = 40 .
Fig. 3. Vorticity. Immediate images of the flow at the time moments: (a) t − t₁ = 0, (b) t − t₁ = 0.6 , (c) t − t₁ = 1.2 , (d) t − t₁ = 1.8. Re = 40 , u₀ = −0.25 , u₁ = 1 .
Fig. 4. The structure of forces acting on the sphere subjected to constant acceleration. Left-side image – Re = 10 (u₀ = 0.5, u₁ = 1,  ∆t = 0.5) , right-side image – Re = 100 (u₀ = −0.1, u₁ = 1, ∆t = 0.01) . The lines designate components of the force: F (gray solid line), F_p  (black dash-and-dot line), F_v  (gray dashed line), markers – data from [13].

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For  citation:  Nuriev  A.N.,  Zaitseva  O.N.,  Yunusova  A.I.  Numerical  investigation  of the history and added mass forces acting on a spherical micro-particle in rectilinear motion in the case of finite Reynolds numbers. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 4, pp. 458–472. (In Russian)

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