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��Combustion, Explosion, and Shock Waves, Vol. 38, No. 3, pp. 322 326, 2002 Calculation of Dust Lifting by a Transient Shock Wave Yu. A. Gosteev1 and A. V. Fedorov2 UDC 534.222.2; 662.612.32 Translated from Fizika Goreniya i Vzryva, Vol. 38, No. 3, pp. 80 84, May June, 2002. Original article submitted May 17, 2001. A mathematical model of lifting of dust particles under the action of transient shock waves is proposed, which takes into account a simultaneous action of the Saffman force and aerodynamic interference on the particle. This model provides an adequate description of the initial stage of lifting of single particles of a dusty layer under the action of shock waves of weak and moderate strength. Satisfactory agreement of numerical and experimental data is reached. It is shown that particle lifting is caused by the action of the Saffman force in the case of weak shock waves (the shock-wave Mach number is less than 1.5) and medium-sized particles (the particle diameter is less than 100 �m) and aerodynamic interference between the particle and the surface in the case of shock waves of moderate strength (the shock-wave Mach number is 2.1 3.3) and large particles (the particle diameter is 200 250 �m). Key words: two-phase flows, shock waves, formation of gas mixtures. The problem of lifting dusty deposits of com- another physical model of the phenomenon, which was bustible disperse materials by transient shock waves has based on the allowance for the repulsion force caused by aroused much interest among researchers. This is not aerodynamic interference of the particle with the surface accidental, since the stage of the formation of a dust air when a shock wave of a rather large amplitude passes mixture plays an important role in further development along the surface. There are also other approaches for of reaction of the cloud of combustible particles, includ- explaining lifting of dust particles, for instance, the al- ing the development toward explosion and detonation lowance for the rotational motion of the particle and regimes of combustion. the associate Magnus force (see [4]), etc. A review of the existing physical conceptions and The role of this or that mechanism of particle lift- mathematical models that describe this phenomenon ing is revealed by considering quantitative relations be- can be found in [1, 2]. The conclusions of Borisov et tween the shock-wave strength, the growth rate of the al. [1] are briefly described as follows. The mechanism boundary layer, the particle size, and other parameters. of dust blowout due to the formation of a series of com- For example, for the case of strong shock waves and pression and rarefaction waves in a dusty layer cannot coarse particles, which do not enter the boundary layer be used to describe the dynamics of mixture formation. for a long time or do not enter it at all, the mechanism At the same time, the initial stage of dust blowout can of lifting caused by aerodynamic interference is more be described using the mechanism of particle motion, important. Therefore, in the present paper, we propose which takes into account the shear flow of the gas in a combined mathematical model for the description of the boundary layer. Later, Volkov et al. [3] proposed lifting of dusty layer particles in the regime of single particles, which simultaneously takes into account the 1 action of the Saffman force and aerodynamic interfer- Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, ence. Novosibirsk 630090; gosteev@itam.nsc.ru. 2 Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, 630090 Novosibirsk; Novosibirsk State University of Architecture and Construction, Novosibirsk 630008; fedorov@ngasu.nsk.ru. 322 0010-5082/02/3803-0322 $27.00 � 2002 Plenum Publishing Corporation Calculation of Dust Lifting by a Transient Shock Wave 323 FORMULATION OF THE PROBLEM the pressure at rest and the ratio of specific heats, Cp is AND GOVERNING EQUATIONS the pressure coefficient in aerodynamic interference of the particle and the wall, and Mmax(M0) is the maxi- We consider a solid particle of spherical shape, mum relative Mach number of the gas flow arising as the which rests on a flat surface. If a normal shock wave ar- shock wave passes near the particle [3]. For Mmax > 1, rives on the plate, its slipping along the surface leads to this quantity determines the possibility of the formation the formation of the boundary layer behind the shock- of a detached shock wave near the particle: wave front. The particle may start to move along the Mmax = 2(M2 - 1){[M2(� - 1) + 2] plate in the gas flow behind the transient shock wave [4]. 0 0 In the present study, however, we ignore particle slip- ping and rolling, assuming, for instance, that the par- � [2�M2 - (� - 1)]}-1/2. 0 ticle is located in a shallow cavity. We assume that The action of the force fint is limited by the distance the main mechanism of particle lifting is the action of �int expressed in particle diameters. We assume that the l the Saffman force fS [5] and the force of aerodynamic particle/surface interaction consists the following mech- interference fint [3], with counteracting forces of aero- anisms: dynamic drag [fa = (fa,x, fa,y)] and gravity (fgr). We (a) the action on the particle of the shock wave attach the origin of the coordinate system (x, y), where caused by reflection of the attached shock wave from y is the vertical coordinate, to the initial position of the the plane; particle and write the equations of motion (b) the lift force arising due to the pressure differ- dup dvp ence on the upper and lower particle surfaces because mp = fx, mp = fy, dt dt of flow deceleration in the region of the lower part of (1) the particle [6]. dxp dyp = up, = vp. For the drag coefficient, we use the dependence [7] dt dt Here xp and yp are the coordinates, up and vp are the 24 4.4 cD = + + 0.42. longitudinal and vertical components of velocity, dp, Rep p Re0.5 mp = �p�d3/6, and �p are the particle diameter, mass, p The initial conditions for system (1) have the fol- and density, respectively. The force acting on the par- lowing form: ticle has the following components: �d2 �g(ug - up)wp,g p t = 0: xp = 0, yp = rp = dp/2, fx = fa,x = cD , (3) 4 2 up = 0, vp = 0. fy = fgr + fa,y + fS + fint, Thus, the particle motion in a known flow field of �d2 �gvpwp,g p the gas behind the transient shock wave with the param- fgr = -mpg, fa,y = - cD , (2) 4 2 eters ug(x, y, t) and �g(x, y, t) is described by a system of ordinary differential equations (1) supplemented by d2 "ug 1/2 p fS = cS�g(ug - up) � (xp, yp) , expressions (2) for the forces acting on the particle and 4 "y the initial conditions (3). �� 0, Mmax(M0) 1; �� d2 p0M2Cp, Mmax(M0) > 1, �� p 0 2 VERIFICATION OF THE MODEL fint = �int; yp/dp l �� 0, Mmax(M0) > 1, �� The mathematical model was verified by numerical �� int. yp/dp > l calculations of particle dynamics on the basis of system Here ug = ug(xp, yp, t) and �g = �g(xp, yp, t) are the (1) (3). longitudinal component of velocity and gas density at Weak Shock Wave. First, we compare the cal- 2 the point of particle location, wp,g = [(u2 - u2) + vp]1/2 culations with the experimental data of [8] in terms g p is the particle velocity relative to the gas, � is the of the dependence of the height of soot-particle lift- kinematic viscosity of the gas, cD = cD(Rep) and cS ing (dp 40 �m and �p = 2900 kg/m3) on the dis- are the coefficients of the drag and Saffman forces, tance from the particle to the front of the transient Rep = wp,gdp/� is the Reynolds number of the par- weak (M0 H" 1.2 1.4) shock wave. Since Merzkirch and ticle, M0 is the shock-wave Mach number, p0 and � are Bracht [8] considered the initial period of dust lifting 324 Gosteev and Fedorov Because of the significant uncertainty in defining the coefficient of the Saffman force cS in the near-wall flow (see, e.g., [10]), the value of this parameter was chosen from the condition of the best agreement be- tween numerical and experimental data for a particle of size dp = 30 �m. Figure 1a shows a comparison of numerical and experimental data for two variants of the flow: 1) uSW = 434 m/sec and u" = 133 m/sec; 2) uSW = 385 m/sec and u" = 64 m/sec. For the first variant, numerical results (curve 3) for the value cS = 32.2 recommended by Merzkirch and Bracht [8] are also plotted. Satisfactory agreement of numerical and experimental results is reached for cS = 160. With increasing shock-wave strength and distance from the particle to the wave front, the accuracy of predicting the lifting height becomes worse. Real conditions imply polydispersity of the parti- cles of the loose layer along which the initiating shock wave moves. Therefore, it is of interest to study the influence of the particle size on the dynamics of its lift- ing (see Fig. 1b, which shows the numerical results for cS = 32.2). It is seen from Fig. 1b that an increase in particle diameter leads to a decrease in the lifting height in the time interval where the shock wave has not moved too far ahead of the particle (xSW - xp < 150 mm). Fig. 1. Height of lifting of a soot particle on its po- Then, small particles (dp 10 �m) relax more rapidly sition with respect to the front of the transient shock to the gas velocity, and their motion with this ve- wave: (a) dp = 30 �m, the solid curves show the cal- locity is quasisteady for some time. Large particles culations for uSW = 434 m/sec and cS = 160 (curve (dp > 10 �m) have a greater relaxation time, which 1), uSW = 385 m/sec and cS = 160 (curve 2), and uSW = 434 m/sec and cS = 32.2 (curve 3); the is responsible for the nonmonotonic behavior of their dashed curve shows the local thickness of the bound- trajectories (see curves 2 4). The total height of parti- ary layer behind the shock wave (uSW = 385 m/sec) cle lifting is also a nonmonotonic function of the parti- and points show the experimental data of [7] for cle size. For particle diameters smaller than H"30 �m, the same shock-wave parameters as (1) and (2); (b) the dependence yp = yp(xSW - xp) is increasing, since effect of the particle size on lifting dynamics for uSW = 385 m/sec, cS = 32.2, and dp = 10 (curve 1), the coarse particles are more inertial. As the particle 20 (curve 2), 30 (curve 3), and 50 �m (curve 4). size further increases, particle deceleration due to an in- crease in its mass starts to be manifested, and the lifting height decreases. The results presented show that nu- from a flat plate, the boundary layer formed was lami- merical data cannot be fitted to experimental results for nar. In this case, the steady flow field of the gas can be cS = 32.2 by changing the particle size. represented in the following form [8, 9]: �y u" sin , 0 y �, ug = 2� u", y > �, TABLE 1 Efficiency of Action of the Saffman Force (uSW = 434 m/sec, cS = 160, and �p = 2900 kg/m3) � = a�[�(uSWt - x)]1/2, a� = 3.64, �g = const. (4) Particle Vertical velocity Here � is the boundary-layer thickness, uSW is the diameter, of the particle leaving shock-wave velocity, and u" is the gas velocity behind �m the boundary layer, m/sec the shock wave. Since we have Mmax < 1 under these 10 7.569 conditions, the detached shock wave near the particle is not formed, and the interference force does not arise 20 5.637 (fint = 0) [possible effect of mechanism (b) of interfer- 30 4.715 ence interaction is ignored]. 50 3.765 Calculation of Dust Lifting by a Transient Shock Wave 325 TABLE 2 Efficiency of Action of the Force of Aerodynamic Interference �int (M0 = 2.7, l = 4.13, and �p = 1200 kg/m3) Particle Vertical velocity diameter, of the particle �m after interaction, m/sec 200 17.652 225 17.663 250 17.668 300 17.681 400 17.699 Fig. 2. Dynamics of lifting of a Plexiglas particle: (a) dp = 225 �m, the solid curves show the calculations for M0 = 2.7 (1) and 2.1 (2); the points refer to the experimental data of [4] for M0 = 2.7 ( ) and 2.1 ( ); Fig. 3. Dynamics of lifting of a bronze particle for (b) curve 3 refers to particle lifting and curve 4 shows M0 = 2.7: the solid curve refers to the numerical data the growth of the local boundary-layer thickness. of the authors for dp = 225 �m and the points show the experimental data of [4]. The values of the vertical velocity of the particle leaving the boundary layer, which are acquired under on a transverse subsonic flow around a cylinder near a the action of the Saffman force, are listed in Table 1. plane and gave the following estimate for the maximum When moving inside the boundary layer smaller parti- thickness H of a slop that ensures the regime of the most cles have a greater vertical velocity. intense aerodynamic interference: H/dp < 4 or (since �int Shock Wave of Moderate Strength. Boiko H = lint - rp) l < 4.5. Certainly, the flow around a and Papyrin [4] gave data on lifting of Plexiglas (�p = sphere is less constrained than the flow around a cylin- 1200 kg/m3) and bronze (�p = 8600 kg/m3) particles by der. In addition, the supersonic flow regime should be the shock wave (M0 = 2.1 3.3). The range of particle taken into account. All these factors may somehow af- �int, diameters was dp = 200 250 �m. fect the experimental value of l but as a whole, quan- In the previous section, we identified the Saffman titative agreement of theoretical estimates and experi- �int force coefficient. Now the free parameters of the model mental data on l is observed. are the pressure coefficient Cp and the maximum dis- The results of trajectory calculations of a Plexi- tance of aerodynamic interaction of the particle with glas particle (dp = 225 �m) for M0 = 2.1 and 2.7 �int the surface l = lint/dp. Following [3], we assume that (Mmax = 1.268) are compared with experimental data Cp = 0.02. A series of trajectory calculations of Plex- in Fig. 2a. Note, better agreement with the experi- iglas particles for M0 = 2.1 (Mmax = 1.017) yielded ment can be reached by taking into account the increase �int an optimal value l = 4.13. To support this result, in the pressure coefficient with increasing shock-wave we note that Petrov [6] generalized experimental data Mach number. Figure 2b shows the time evolution of 326 Gosteev and Fedorov the vertical coordinate of the particle (curve 3) and lo- REFERENCES cal boundary-layer thickness �(xp, t) (curve 4). It is seen 1. A. A. Borisov, B. E. Gel fand, and S. A. Tsyganov, that a coarse particle during its lifting does not enter Mixing behind shock waves and detonation in dust the boundary layer at all ( jumps out of it), i.e., the gas mixtures, in: Book of Papers of the First Int. Coll. Saffman force does not act on the particle in this case on Explosibility of Industrial Dusts, Part 2, November (fS = 0). 8 10, Baranow (1984), pp. 137 161. The vertical velocity of the particle, which is ac- 2. A. V. Fedorov, T. A. Khmel , N. N. Fedorova, et al., quired under the action of the aerodynamic interference Review of experimental studies and analysis of existing force, are listed in Table 2. An increase in particle size mathematical models of mixing processes for air coal leads to a weak increase in lifting velocity. The rea- dust systems, Report of Inst. Theor. Appl. Mech., Sib. son is that, though a larger particle has a greater mass, Div., Russian Acad. Sci., Novosibirsk (1999). it experiences the action of the interference force for 3. V. F. Volkov, A. V. Fedorov, and V. M. Fomin, Prob- a longer time, since lint increases in proportion to the lem of the interaction between a supersonic flow and particle size. a cloud of particles, J. Appl. Mech. Tech. Phys., 35, Satisfactory agreement is obtained for the growth No. 6, 832 836 (1994). of lifting height of a bronze particle (dp = 225 �m) 4. V. M. Boiko and A. N. Papyrin, Dynamics of the for- behind a transient shock wave with a Mach number of mation of a gas suspension behind a shock wave sliding 2.7 in the time interval up to 200 �sec (Fig. 3). The over the surface of a loose material, Combust. Expl. experimental data shown in Fig. 3 correspond to the Shock Waves, 23, No. 2, 231 235 (1987). maximum lifting height of particles of a dusty layer. 5. P. G. Saffman, The lift on a small sphere in a slow shear flow, J. Fluid Mech., 22, 383 400 (1965). 6. K. P. Petrov, Aerodynamics of Bodies of Simplest Shapes CONCLUSIONS [in Russian], Faktorial, Moscow (1998). 7. L. E. Sternin, B. I. Maslov, A. A. Shraiber, et al., Two- " A mathematical model based on the Saffman and Phase Mono- and Polydisperse Gas Particle Flows [in aerodynamic interference forces is used to describe lift- Russian], Mashinostroenie, Moscow (1980). ing of dust particles in the flow field of a gas within the 8. W. Merzkirch and K. Bracht, The erosion of dust by a Lagrangian approach. shock wave in air: Initial stages with laminar flow, Int. " The initial stage of lifting of single particles of J. Multiphase Flow, 4, 89 95 (1978). a dusty layer under the action of shock waves of weak 9. G. Schlichting, Boundary Layer Theory, McGraw-Hill, and moderate strength is adequately described using New York (1968). this model. 10. L. B. Gavin and A. A. Shraiber, Turbulent gas particle " It is shown that particle lifting is caused by the flows, in: Itogi Nauki Tekh., Ser. Mekh. Zhidk. Gaza, Saffman force in the case of weak shock waves and 25, 90 182 (1991). medium-size particles and by the force of aerodynamic interference between the particle and the surface in the case of medium-strength shock waves and large parti- cles. This work was supported by the Russian Foun- dation for Fundamental Research (Grant No. 99 01 00587), INTAS OPEN (Grant No. 97-2027), ISTC (Grant No. 612), and Mekhanika (Grant No. E00-4.0- 90).

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