Modeling of the unsteady force for shock-particle interaction. (English) Zbl 1255.76062
Summary: The interaction between a particle and a shock wave leads to unsteady forces that can be an order of magnitude larger than the quasi-steady force in the flow field behind the shock wave. Simple models for the unsteady force have so far not been proposed because of the complicated flow field during the interaction. Here, a simple model is presented based on the work of M. Parmar et al. [Phil. Trans. R. Soc, A 366, 2161–2175 (2008; Zbl 1256.76072)]. Comparisons with experimental and computational data for both stationary spheres and spheres set in motion by shock waves show good agreement in terms of the magnitude of the peak and the duration of the unsteady force.
Citations:
Zbl 1256.76072References:
| [1] | Bagchi P., Balachandar S.: Inertial and viscous forces on a rigid sphere in straining flows at moderate Reynolds numbers. J. Fluid Mech. 481, 105–148 (2003) · Zbl 1064.76022 · doi:10.1017/S002211200300380X |
| [2] | Bredin M.S., Skews B.W.: Drag measurements in unsteady compressible flow. Part 1: an unsteady flow facility and stress wave balance. R&D J. South Afr. Inst. Mech. Eng. 23, 3–12 (2007) |
| [3] | Britan A., Elperin T., Igra O., Jiang J.P.: Acceleration of a sphere behind planar shock-waves. Exp. Fluids 20(2), 84–90 (1995) · doi:10.1007/BF01061585 |
| [4] | Bryson A.E., Gross R.W.F.: Diffraction of strong shocks by cones, cylinders and spheres. J. Fluid Mech. 10(1), 1–16 (1961) · Zbl 0100.22103 · doi:10.1017/S0022112061000019 |
| [5] | Carrier G.F.: Shock waves in a dusty gas. J. Fluid Mech. 4(4), 376–382 (1958) · Zbl 0082.40305 · doi:10.1017/S0022112058000513 |
| [6] | Chang E.J., Maxey M.R.: Unsteady-flow about a sphere at low to moderate Reynolds-number. 2. accelerated motion. J. Fluid Mech. 303, 133–153 (1995) · Zbl 0938.76072 · doi:10.1017/S0022112095004204 |
| [7] | Clift, R., Gauvin, W.H.: The motion of particles in turbulent gas streams. In: Proc. Chemeca ’70, vol. 1, pp. 14–28 (1970) |
| [8] | Devals C., Jourdan G., Estivalezes J.L., Meshkov E.E., Houas L.: Shock tube spherical particle accelerating study for drag coefficient determination. Shock Waves 12(4), 325–331 (2003) · doi:10.1007/s00193-002-0172-z |
| [9] | Eames I., Hunt J.C.R.: Forces on bodies moving unsteadily in rapidly compressed flows. J. Fluid Mech. 505, 349–364 (2004) · Zbl 1067.76077 · doi:10.1017/S0022112004008535 |
| [10] | Gatignol R.: The Faxén formulas for a rigid particle in an unsteady non-uniform stokes-flow. J. Mec. Theor. Appl. 2(2), 143–160 (1983) · Zbl 0544.76032 |
| [11] | Gilbert M., Davis L., Altman D.: Velocity lag of particles in linearly accelerated combustion gases. Jet Propulsion 25(1), 26–30 (1955) · doi:10.2514/8.6578 |
| [12] | Haselbacher, A.: A WENO reconstruction algorithm for unstructured grids based on explicit stencil construction. AIAA Paper 2005-0879 (2005) |
| [13] | Haselbacher A., Balachandar S., Kieffer S.W.: Open-ended shock tube flows: influence of pressure ratio and diaphragm position. AIAA J. 45(8), 1917–1929 (2007) · doi:10.2514/1.23081 |
| [14] | Igra O., Takayama K.: Shock-tube study of the drag coefficient of a sphere in a nonstationary flow. Proc. R. Soc. Lond. A 442(1915), 231–247 (1993) · doi:10.1098/rspa.1993.0102 |
| [15] | Ingebo, R.D.: Drag coefficient for droplets and solid spheres in clouds accelerated in air streams. NACA Tech. Note 3762 (1956) |
| [16] | Jourdan G., Houas L., Igra O., Estivalezes J.L., Devals C., Meshkov E.E.: Drag coefficient of a sphere in a non-stationary flow: new results. Proc. R. Soc. A 463(2088), 3323–3345 (2007) · doi:10.1098/rspa.2007.0058 |
| [17] | Kriebel A.R.: Analysis of normal shock waves in particle laden gas. J. Basic Eng. 84(4), 655–665 (1964) · doi:10.1115/1.3655914 |
| [18] | Landau L.D., Lifshitz E.M.: Fluid Mechanics, 2nd edn. Butterworth-Heinemann, London (1987) · Zbl 0146.22405 |
| [19] | Lock G.D.: Experimental measurements in a dusty-gas shock-tube. Int. J. Multiphase Flow 20(1), 81–98 (1994) · Zbl 1134.76597 · doi:10.1016/0301-9322(94)90007-8 |
| [20] | Longhorn A.L.: The unsteady, subsonic motion of a sphere in a compressible inviscid fluid. Q. J. Mech. Appl. Math. 5(1), 64–81 (1952) · Zbl 0046.19205 · doi:10.1093/qjmam/5.1.64 |
| [21] | Loth E.: Compressibility and rarefaction effects on drag of a spherical particle. AIAA J. 46(9), 2219–2228 (2008) · doi:10.2514/1.28943 |
| [22] | Love A.E.H.: Some illustrations of modes of decay of vibratory motions. Proc. Lond. Math. Soc. 2, 88–113 (1905) · JFM 35.0820.01 · doi:10.1112/plms/s2-2.1.88 |
| [23] | Magnaudet J., Eames I.: The motion of high-Reynolds-mumber bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32, 659–708 (2000) · Zbl 0989.76082 · doi:10.1146/annurev.fluid.32.1.659 |
| [24] | Maxey M.R., Riley J.J.: Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26(4), 883–889 (1983) · Zbl 0538.76031 · doi:10.1063/1.864230 |
| [25] | Mei R.W.: Flow due to an oscillating sphere and an expression for unsteady drag on the sphere at finite Reynolds-number. J. Fluid Mech. 270, 133–174 (1994) · Zbl 0818.76012 · doi:10.1017/S0022112094004222 |
| [26] | Mei R.W., Adrian R.J.: 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 (1992) · Zbl 0747.76038 · doi:10.1017/S0022112092003434 |
| [27] | Miles J.W.: On virtual mass and transient motion in subsonic compressible flow. Q. J. Mech. Appl. Math. 4(4), 388–400 (1951) · Zbl 0044.21703 · doi:10.1093/qjmam/4.4.388 |
| [28] | Mougin G., Magnaudet J.: The generalized Kirchhoff equations and their application to the interaction between a rigid body and an arbitrary time-dependent viscous flow. Int. J. Multiphase Flow. 28(11), 1837–1851 (2002) · Zbl 1137.76687 · doi:10.1016/S0301-9322(02)00078-2 |
| [29] | Parmar M., Haselbacher A., Balachandar S.: On the unsteady inviscid force on cylinders and spheres in subcritical compressible flow. Phil. Trans. R Soc. A 366(1873), 2161–2175 (2008) · Zbl 1256.76072 · doi:10.1098/rsta.2008.0027 |
| [30] | Rodriguez, G., Grandeboeuf, P., Khelifi, M., Haas, J.F.: Drag coefficient measurement of spheres in a vertical shock tube and numerical simulation. In: Brun, R., Dumitrescu, L. (eds.) Shock Waves @ Marseille III, pp. 43–48. Springer, Heidelberg (1995) |
| [31] | Rudinger G.: Some properties of shock relaxation in gas flows carrying small particles. Phys. Fluids 7(5), 658–663 (1964) · doi:10.1063/1.1711265 |
| [32] | Rudinger G.: Effective drag coefficient for gas-particle flow in shock tubes. J. Basic Eng. 92(1), 165–172 (1970) · doi:10.1115/1.3424925 |
| [33] | Rudinger, G.: Fundamentals of Gas-Particle Flow. Handbook of Powder Technology, vol. 2. Elsevier Scientific Publishing Co., Amsterdam (1980) |
| [34] | Schiller L., Naumann A.: Über die grundlegenden Kräfte bei der Schwerkraftaufbereitung. Zeitschrift des Vereines Deutscher Ingenieure 77, 318–320 (1933) |
| [35] | Selberg B.P., Nicholls J.A.: Drag coefficient of small spherical particles. AIAA J 6(3), 401–408 (1968) · doi:10.2514/3.4513 |
| [36] | Shapiro, A.H.: The dynamics and thermodynamics of compressible fluid flow. The Ronald Press Company, New York (1953) · Zbl 0053.00803 |
| [37] | Skews B.W., Bredin M.S., Efune M.: Drag measurements in unsteady compressible flow. Part 2: shock wave loading of spheres and cones. R&D J. South Afr. Inst. Mech. Eng. 23, 13–19 (2007) |
| [38] | Sommerfeld M.: The unsteadiness of shock-waves propagating through gas-particle mixtures. Exp. Fluids 3(4), 197–206 (1985) · doi:10.1007/BF00265101 |
| [39] | Soo S.L.: Gas-dynamic processes involving suspended solids. AIChE J. 7(3), 384–391 (1961) · doi:10.1002/aic.690070308 |
| [40] | Sun M., Saito T., Takayama K., Tanno H.: Unsteady drag on a sphere by shock wave loading. Shock Waves 14(1–2), 3–9 (2005) · Zbl 1178.76222 · doi:10.1007/s00193-004-0235-4 |
| [41] | Suzuki T., Sakamura Y., Igra O., Adachi T., Kobayashi S., Kotani A., Funawatashi Y.: Shock tube study of particles’ motion behind a planar shock wave. Meas. Sci. Tech. 16(12), 2431–2436 (2005) · doi:10.1088/0957-0233/16/12/005 |
| [42] | Tanno H., Itoh K., Saito T., Abe A., Takayama K.: Interaction of a shock with a sphere suspended in a vertical shock tube. Shock Waves 13(3), 191–200 (2003) · doi:10.1007/s00193-003-0209-y |
| [43] | Tanno H., Komuro T., Takahashi M., Takayama K., Ojima H., Onaya S.: Unsteady force measurement technique in shock tubes. Rev. Sci. Instrum. 75(2), 532–536 (2004) · doi:10.1063/1.1641156 |
| [44] | Taylor, G.I.: The motion of a body in water when subjected to a sudden impulse. In: Batchelor, G. (ed.) The Scientific Papers of G.I. Taylor, vol. 3, pp. 306–308. The Cambridge University Press, New York (1942) |
| [45] | Wakaba L., Balachandar S.: On the added mass force at finite Reynolds and acceleration numbers. Theor. Comput. Fluid Dyn. 21(2), 147–153 (2007) · Zbl 1161.76462 · doi:10.1007/s00162-007-0042-5 |
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