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Relaxation drag history of shock accelerated microparticles

Published online by Cambridge University Press:  21 June 2017

Ankur D. Bordoloi Open the ORCID record for Ankur D. Bordoloi [Opens in a new window] ,
Adam A. Martinez  and
Katherine Prestridge Open the ORCID record for Katherine Prestridge [Opens in a new window]
Ankur D. Bordoloi*
Affiliation:
Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Adam A. Martinez
Affiliation:
Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Katherine Prestridge
Affiliation:
Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
*
†Email address for correspondence: ankur@lanl.gov
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Abstract

Experimental measurements of the displacements of shock accelerated microparticles from shortly after shock interaction to the particle relaxation time show time-dependent drag coefficients ($C_{D}$) that are much higher than those predicted by quasi-steady and unsteady drag models. Nylon particles with mean diameter of $4~\unicode[STIX]{x03BC}\text{m}$, accelerated by one-dimensional normal shocks (Mach number $M_{s}=1.2$, 1.3 and 1.4), have measured $C_{D}$ values that follow a power-law behaviour. The drag is a function of the time-dependent Knudsen number, $Kn^{\ast }=M_{s}/Re_{p}$, where the particle Reynolds number ($Re_{p}$) is calculated using the time-dependent slip velocity. Some portion of the drag can be attributed to quasi-steady forces, but the total drag cannot be predicted by current unsteady force models that are based on the Basset–Boussinesq–Oseen equation and pressure drag. The largest contribution to the total drag is the unsteady component ($C_{D,us}$) until the particle attains $Kn^{\ast }\approx 0.5{-}1.0$, then the unsteady contribution decays. The quasi-steady component ($C_{D,qs}$) increases almost linearly with $Kn^{\ast }$, intersects the $C_{D,us}$ at $Kn^{\ast }\approx 2$ and becomes the primary contributor to the drag towards the end of the relaxation zone as $Re_{p}\rightarrow 0$. There are currently no analytical models that are able to predict the nonlinear behaviour of the shock accelerated particles during the relaxation phase of the flow.

JFM classification

Aerodynamics: High-speed flow Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Compressible Flows: Shock waves
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 823 , 25 July 2017 , R4
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Copyright
© 2017 Cambridge University Press 

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References

Anderson, J. D. 1990 Modern Compressible Flow, 2nd edn. McGraw-Hill.Google Scholar
Boiko, V. M., Kiselev, V. P., Kiselev, S. P., Papyrin, A. N., Poplavsky, S. V. & Fomin, V. M. 1997 Shock wave interaction with a cloud of particles. Shock Waves 7, 275–285.Google Scholar
Christensen, K. T. & Adrian, R. J. 2002 Measurement of instantaneous Eulerian acceleration fields by particle image accelerometry. Exp. Fluids 33, 759–769.Google Scholar
Clift, R. & Gauvin, W. H. 1970 The motion of particles in turbulent gas streams. Proc. Chemeca’70 1, 14–28.Google Scholar
Geng, J. H. & Groenig, H. 2000 Dust suspensions accelerated by shock waves. Exp. Fluids 28, 360–367.CrossRefGoogle Scholar
Gore, R. & Crowe, C. 1989 Effect of particle size on modulating turbulent intensity. Intl J. Multiphase Flow 15 (2), 279–285.Google Scholar
Henderson, C. 1976 Drag coefficients of spheres in continuum and rarefied flows. AIAA J. 14 (6), 707–708.CrossRefGoogle Scholar
Igra, O. & Takayama, K. 1993 Shock tube study of the drag coefficient of a sphere in a non-stationary flow. Proc. R. Soc. Lond. A 442, 231–247.Google Scholar
Jourdan, G., Houas, L., Igra, O., Estivalezes, J. L., Devals, C. & Meshkov, E. E. 2007 Drag coefficient of a sphere in a non-stationary flow: new results. Proc. R. Soc. Lond. A 463, 3323–3345.Google Scholar
Kulick, J. D., Fessler, J. R. & Eaton, J. K. 1994 Particle response to turbulence modification in fully developed channel flow. J. Fluid Mech. 277, 109.CrossRefGoogle Scholar
Longhorn, A. L. 1952 The unsteady, subsonic motion of a sphere in a compressible inviscid fluid. Q. J. Mech. Appl. Maths 5, 64–81.Google Scholar
Loth, E. 2008 Compressibility and rarefaction effects on drag of a spherical particle. AIAA J. 46 (9), 2219–2228.Google Scholar
Magnaudet, J. & Eames, I. 2000 The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32, 659–708.CrossRefGoogle Scholar
Martinez, A. A., Orlicz, G. C. & Prestridge, K. P. 2015 A new experiment to measure shocked particle drag using multi-pulse particle image velocimetry and particle tracking. Exp. Fluids 56, 1854.Google Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883.Google Scholar
Mei, R. & Adrian, R. J. 1992 Flow past a sphere with an oscillation in the free-stream velocity and unsteady drag at finite Reynolds number. J. Fluid Mech. 237, 323–341.Google Scholar
Mei, R., Lawrencce, J. & Adrian, R. J. 1991 Unsteady drag on a sphere at finite Reynolds number with small fluctuations in the free-stream velocity. J. Fluid Mech. 233, 613–631.Google Scholar
Mejia-Alvarez, R., Wilson, B., Leftwich, M. C., Martinez, A. A. & Prestridge, K. P. 2015 Design of a fast diaphragmless shock tube driver. Shock Waves 25 (6), 635–650.CrossRefGoogle Scholar
Miles, J. W. 1951 On virtual mass and transient motion in subsonic compressible flow. Q. J. Mech. Appl. Maths 4, 388–400.Google Scholar
Miura, H. & Glass, I. I. 1983 On the passage of a shock wave through a dusty-gas layer. Proc. R. Soc. Lond. A 385, 85–105.Google Scholar
Murakama, T. & Ishikawa, M. 1978 Holographic measurements of velocity distribution of particles accelerated by a shock wave. In Proceedings of the 13th International Congress on High Speed Photography and Photonics, Proc. SPIE 0189, pp. 326–329.Google Scholar
Parmar, M., Haselbacher, A. & Balachandar, S. 2008 On the unsteady inviscid force on cylinders and spheres in subcritical compressible flow. Phil. Trans. R. Soc. Lond. A 366, 2161–2175.Google ScholarPubMed
Parmar, M., Haselbacher, A. & Balachandar, S. 2009 Modeling of the unsteady force for shock–particle interaction. Shock Waves 19, 317–329.Google Scholar
Parmar, M., Haselbacher, A. & Balachandar, S. 2010 Improved drag correlation for spheres and application to shock-tube experiments. AIAA J. 48, 1273–1276.Google Scholar
Parmar, M., Haselbacher, A. & Balachandar, S. 2011 Generalized Basset–Boussinesq–Oseen equation for unsteady forces on a sphere in a compressible flow. Phys. Rev. Lett. 106, 084501.Google Scholar
Rudinger, G. 1964 Some properties of shock relaxation in gas flows carrying small particles. Phys. Fluids 7 (5), 658–663.Google Scholar
Rudinger, G. 1970 Effective drag coefficient for gas–particle flow in shock tubes. Trans. ASME J. Basic Engng 92, 165–172.Google Scholar
Saito, T., Saba, M., Sun, M. & Takayama, K. 2007 The effect of an unsteady drag force on the structure of a non-equilibrium region behind a shock wave in a gas–particle mixture. Shock Waves 17, 255–262.CrossRefGoogle Scholar
Settles, G. S. 2006 Schlieren and Shadowgraph Techniques. Springer.Google Scholar
Smolders, H. J. & van Dongen, M. E. H. 1992 Shock wave structure in a mixture of gas, vapour and droplets. Shock Waves.Google Scholar
Sommerfeld, M. 1985 The unsteadiness of shock waves propagating through gas–particle mixtures. Exp. Fluids 3, 197–206.Google Scholar
Strecker, J. J. F. & Roth, P. 1994 Particle breakup in shock waves studied by single particle light scattering. Part. Part. Syst. Charact. 11, 222–226.Google Scholar
Sun, M., Saito, T., Takayama, K. & Tanno, H. 2005 Unsteady drag on a sphere by shock wave loading. Shock Waves 14 (1), 3–9.Google Scholar
Tanaka, T. & Eaton, J. K. 2010 Sub-Kolmogorov resolution particle image velocimetry measurements of particle-laden forced turbulence. J. Fluid Mech. 643, 177–206.CrossRefGoogle Scholar
Wagner, J., Beresh, S., Kearney, S., Pruett, B. & Wright, E. 2012 Shock tube investigation of quasi-steady drag in shock–particle interactions. Phys. Fluids 24, 123301.Google Scholar