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
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), Fp (black dash-and-dot line), Fv (gray dashed line), markers – data from [13].
References
1. Van Overbrüggen T., Klaas M., Soria J., Schröder W. Experimental analysis of particle sizes for PIV measurements. Meas. Sci. Technol., 2016, vol. 27, no. 9, art. 094009, pp. 1–10. doi: 10.1088/0957-0233/27/9/094009.
2. Mei R., Adrian R. Flow past a sphere with an oscillation in the free-stream velocity, and unsteady drag at finite Reynolds number. J. Fluid Mech., 1992, vol. 237, pp. 323–341. doi: 10.1017/S0022112092003434.
3. Mei R. History force on a sphere due to a step change in the free-stream velocity. Int. J. Multiphase Flow, 1993, vol. 19, no. 3, pp. 509–525. doi: 10.1016/0301-9322(93)90064-2.
4. Lawrence C.J., Mei R. Long-time behaviour of the drag on a body in impulsive motion. J. Fluid Mech., 1995, vol. 283, pp. 301–327. doi: 10.1017/S0022112095002333.
5. Mei R. Velocity fidelity of flow tracer particles. Exp. Fluids, 1996, vol. 22, no. 1, pp. 1–13. doi: 10.1007/BF01893300.
6. Lighthill M.J. On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Math., 1952, vol. 5, no. 2, pp. 109–118. doi: 10.1002/cpa.3160050201.
7. Egorov A.G., Zakharova O.S. The optimal quasi-stationary motion of a vibration-driven robot in a viscous medium. Russ. Math., 2012, vol. 56, no. 2, pp. 50–55. doi: 10.3103/S1066369X12020065.
8. Egorov A.G., Zakharova O.S. The energy-optimal motion of a vibration-driven robot in a medium with a inherited law of resistance. J. Comput. Syst. Sci. Int., 2015, vol. 54, no. 3, pp. 495–503. doi: 10.1134/S1064230715030065.
9. Childress S., Spagnolie S.E., Tokieda T.A. A bug on a raft: Recoil locomotion in a viscous fluid. J. Fluid Mech., 2011, vol. 669, pp. 527–556. doi: 10.1017/S002211201000515X.
10. Nuriev A., Zakharova O. The optimal control of a multi-mass vibration propulsion system in a viscous incompressible fluid. Proc. 7th Eur. Congr. on Computational Methods in Applied Sciences and Engineering, 2016, vol. 4, pp. 7121–7129. doi: 10.7712/100016.2322.11098.
11. Basset A.B. A Treatise on Hydrodynamics, with Numerous Examples. Vol. 2. Cambridge, Deighton, Bell and Co., 1888. 368 p.
12. Chang E.J., Maxey M.R. Unsteady flow about a sphere at low to moderate Reynolds number. Part 1. Oscillatory motion. J. Fluid Mech., 1994, vol. 277, pp. 347–379. doi: 10.1017/S002211209400279X.
13. Chang E.J., Maxey M.R. Unsteady flow about a sphere at low to moderate Reynolds number. Part 2. Accelerated motion. J. Fluid Mech., 1995, vol. 303, pp. 133–153. doi: 10.1017/S0022112095004204.
14. Nuriev A.N., Zaitseva O.N. Solving the problem of oscillatory motion of a cylinder in a viscous medium using the OpenFOAM package. Vestn. Kazan. Tekhnol. Univ., 2013, vol. 16, no. 8, pp. 116–123. (In Russian)
15. Egorov A.G., Kamalutdinov A.M., Nuriev A.N., Paimushin V.N. Theoretical-experimental method for determining the parameters of damping based on the study of damped flexural vibrations of test specimens 2. Aerodynamic component of damping. Mech. Compos. Mater., 2014, vol. 50, no. 3, pp. 267–278. doi: 10.1007/s11029-014-9413-3.
16. Nuriev A.N., Zakharova O.S. Numerical simulation of the motion of a wedge-shaped two-mass vibration-driven robot in a viscous fluid. Vychisl. Mekh. Sploshnykh Sred, 2016, vol. 9, no. 1, pp. 5–15. doi: 10.7242/1999-6691/2016.9.1.1. (In Russian)
17. Jasak H., Weller H.G., Gosman A.D. High resolution NVD differencing scheme for arbitrarily unstructured meshes. Int. J. Numer. Methods Fluids, 1999, vol. 31, no. 2, pp. 431–449.
18. Jasak H. Error analysis and estimation for the finite volume method with applications to fluid flows. PhD Thesis. London, Univ. of London, Imp. Coll., 1996. 394 p.
19. Issa R.I. Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys., 1986, vol. 62, no. 1, pp. 40–65. doi: 10.1016/0021-9991(86)90099-9.
20. Behrens T. Technical Report. OpenFOAM's Basic Solvers for Linear Systems of Equations: Solvers, Preconditioners, Smoothers. Denmark, Techn. Univ. of Denmark, 2009. 18 p.
21. Le Clair B.P., Hamielec A.E., Pruppracher H.R. A numerical study of the drag on a sphere at low and intermediate Reynolds numbers. J. Atmos. Sci., 1970, vol. 27, pp. 308–315.
22. Dennis S.C.R., Walker J.D.A. Calculation of the steady flow past a sphere at low and moderate Reynolds numbers. J. Fluid Mech., 1971, vol. 48, no. 4, pp. 771–789. doi: 10.1017/S0022112071001848.
23. Johnson T., Patel V. Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech., 1999, vol. 378, pp. 19–70. doi: 10.1017/S0022112098003206.
24. Schlichting H. Boundary Layer Theory. New York, McGraw-Hill, 1979. 817 p.
25. Dennis S.C.R., Walker J.D.A. Numerical solutions for time-dependent flow past an impulsively started sphere. Phys. Fluids, 1972, vol. 15, no. 4, pp. 517–525. doi: 10.1063/1.1693943.
26. Rivero M., Magnaudet J., Fabre J. Quelques resultants nouveaux concernant les forces exercees sur une inclusion spherique par en icoulement accelkre. C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers., 1991, vol. 312, no. 13, pp. 1499–1506. (In French)
27. Wakaba L., Balachandar S. On the added mass force at finite Reynolds and acceleration numbers. Theor. Comput. Fluid Dyn., 2007, vol. 21, no. 2, pp. 147–153. doi: 10.1007/s00162-007-0042-5.
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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