c Pleiades Publishing, Ltd., 2016. ISSN 0010-5082, Combustion, Explosion, and Shock Waves, 2016, Vol. 52, No. 2, pp. 207–218. c T.A. Khmel’, A.V. Fedorov. Original Russian Text
Effect of Collision Dynamics of Particles on the Processes of Shock Wave Dispersion T. A. Khmel’a and A. V. Fedorova
UDC 532.529+532.5:544.3
Published in Fizika Goreniya i Vzryva, Vol. 52, No. 2, pp. 93–105, March–April, 2016. Original article submitted December 4, 2014.
Abstract: Based on numerical simulations of two-dimensional unsteady flows of gas suspensions, the contribution of particle collisions to dispersion processes during interaction of shock waves with dense dust layers is analyzed. A model of collision dynamics of the two-phase medium based on molecular-kinetic approaches is used. The model is tested by using a problem of a shock wave passing along a dense layer of particles; the model predictions are found to agree well with available experimental data. The problem of interaction of a blast wave with a dense layer on a flat surface is also considered. A comparative analysis of various mechanisms acting on particles and the influence of the initial parameters of the layer on the particle lifting dynamics is performed. A weak effect of the Saffman force and inhomogeneity of the layer surface (waviness) and a significant effect of the Magnus force on dispersion of the layer directly behind the shock wave are demonstrated. In some cases, the contribution of the particle collision dynamics is found to be comparable with the Magnus force effect. Dust lifting due to the development of the Kelvin–Helmholtz instability occurs at late stages of the process. Keywords: gas suspension, shock wave, dispersion, collision dynamics, mathematical modeling. DOI: 10.1134/S0010508216020118
INTRODUCTION Investigations of dispersion of particles under the action of shock waves is primarily associated with various issues of formation of explosive dust clouds and various related problems, such as the dust content in the medium, agglomeration of particles, and conditions of ignition of dust suspensions of reactive particles. A fairly detailed review of experimental and theoretical investigations of mixing in shock wave and detonation processes can be found in [1]. Various aspects of shock wave interaction with dust-laden systems were considered in [2–4]. A typical applied problem is sliding of the shock wave along a dusty layer. For gas media with a low dust content, the main factor of layer expansion is interaction of shock wave structures: refraction and reflection [5, 6], as well as the developa
Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia;
[email protected].
ment of the Kelvin–Helmholtz instability in the surface layer [6]. A possible mechanism associated with generation of a repulsive force in a supersonic flow of closely located bodies (particles) due to aerodynamic interference was discussed in [7]. The Saffman force acting on the particle in a gradient flow field behind the shock wave was taken into account in modeling lifting of individual particles in [8] and lifting of a suspension of particles in [9, 10]. In [11–13], the governing factor was assumed to be the Magnus force arising due to the development of rotational motion of particles, resulting in particle lifting and layer dispersion. Rotation of particles may be induced both by their collisions with an obstacle or other particles [11] and by the action of the crossflow gradient of the gas velocity in the upper layer of the dust suspension [12]. In [13], the Magnus effect was associated with rotation of particles in the course of the development of their random motion and collisions, which was numerically simulated by means of introducing some additional randomization of parameters in computational cells.
c 2016 by Pleiades Publishing, Ltd. 0010-5082/16/5202-0207
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208 One of the factors of expansion of dense dust layers is multiple collisions of particles with each other. It was argued [14] that they play the governing role in expansion of the layer due to the shock wave passing over this layer. The discrete model took into account particle collision with each other and with a rough solid wall, but the shock wave structures inside the layer were not obtained. Collision effects were considered in [9, 10] within the framework of continuum models, which took into account random motion and intrinsic pressure in the particle phase. However, the Magnus force was ignored, and the contribution of collisions among other factors of the dispersion process was not analyzed. Klemens et al. [15] performed an experimental and theoretical study of dust lifting behind the shock wave. They used a continuum-discrete approach, which provided an accurate description of particle collisions with each other. However, the model included the vertical velocity of particles on the layer surface, which was prescribed in accordance with experimental data. Thus, though publications provide results on dust lifting with the use of approaches that take into account particle collisions, the influence of collision dynamics on dispersion of dust layers and cloud formation was not adequately studied, and there is still significant interest to such investigations. The main principles of the collision dynamics description within the framework of mechanics of continuous media were described in [16]. Models of granular media have been recently developed, where particle collisions are described by molecular-kinetic approaches within the framework of mechanics of continuous media [17]. Goldshtein and Shapiro [18] derived the Euler equations for an isolated granular medium with allowance for the nonideal character of collisions of inelastic rough particles. Based on the concept of [18], Fedorov and Khmel’ [19, 20] developed a model for a flow of a two-phase mixture of a gas and fine solid particles with allowance for finite volume concentrations and different properties of particle elasticity and roughness. A class of media for which the model could be presented in a divergent form was found in [20], which made it possible to obtain conditions on shock waves and a classification of strong discontinuities. Two types of combined discontinuities were identified for a two-phase gas–particle mixture with collisions: without and with generation of collision energy on the shock wave. The amplitude of the volume concentration on the collisioninduced shock wave is uniquely determined by the initial value and parameters of restitution and roughness and is independent of the wave propagation velocity. Conditions of existence of shock wave structures of different types were obtained. This model was used in [21]
Khmel’ and Fedorov to analyze shock wave and detonation processes in gas suspensions. Based on solving the Cauchy problem for unsteady flows, steady solutions corresponding to the above-mentioned types of steady waves for ideal and nonideal collisions were obtained. It was also shown [20] that the model predicts data that agree with the experimental data [22] on particle spreading from a high-pressure chamber, which begins after diaphragm breakdown (data on the velocity of sound in the particle phase as a function of the particle size, particle concentration, and pressure in the chamber). In the present work, we use the two-phase medium model [19, 20] based on molecular-kinetic approaches to describe particle collisions and perform a numerical study of two-dimensional unsteady flows of gas suspensions with high-density layers of particles with a finite volume concentration. The model and numerical method are tested by using a problem of propagation of a sustained shock wave along a dense layer of particles under conditions corresponding to the experiments [10]. Further we consider a problem of interaction of a blast wave with a dense layer of particles lying on a flat surface. A comparative analysis of various factors of particle lifting and cloud formation is performed: Saffman force, Magnus force, layer surface shape, development of the Kelvin–Helmholtz instability, random motion of particles, and their collisions. The goal of the present study is to estimate the contribution of particle collision dynamics to the processes of shock wave dispersion of dust suspensions.
COLLISION MODEL OF A TWO-PHASE MEDIUM FOR THE DESCRIPTION OF DYNAMICS OF GAS SUSPENSIONS OF INCOMPRESSIBLE PARTICLES The constitutive equations within the framework of mechanics of interpenetrating continua follow from the mass, momentum, and energy conservation equations for the mixture as a whole. The Euler equations for two-dimensional flows of a nonreacting medium have the following form: ∂F ∂G ∂W + + = Γ; ∂t ∂x ∂y F1 W1 , F = , W = W2 F2 ˜2 G1 −Γ G= , Γ= ; G2 Γ2
(1)
(2)
Effect of Collision Dynamics of Particles on the Processes of Shock Wave Dispersion 209 ⎛ ⎞ ρ1 the phases, and η < 1 is the parameter of nonideal col⎜ ρ1 u 1 ⎟ lisions [20]; the subscripts 1 and 2 refer to the gas phase ⎜ ⎟ , W1 = ⎝ ρ1 v1 ⎠ and particles, respectively. The relations for the interρ1 E1 phase interaction processes (variables f and q) depend ⎛ ⎞ on the proceeding processes and volume concentration ρ1 u 1 of particles. The interphase interaction processes in⎜ m1 p1 + ρ1 u21 ⎟ ⎟, F1 = ⎜ (3) volve the gravity force, the Saffman force [8, 10], and ⎝ ρ1 u1 v1 ⎠ the Magnus force [12]: ρ1 u1 E1 + m1 p1 u1 3m2 ρ11 ⎛ ⎞ cD |u1 − u2 |(u1 − u2 ) f= ρ1 v1 4d ⎜ ρ1 u1 v1 ⎟ ⎟; G1 = ⎜ (7) − g + f Saffman + f Magnus , ⎝ m1 p1 + ρ1 v12 ⎠ 3ρ1 m2 ρ1 v1 E1 + m1 p1 v1 [(u1 − u2 ) × rot(u1 )], (8) f Magnus = KMagnus 4π ⎞ ⎛ ρ2 fSaffman,y ⎟ ⎜
⎜ ρ2 u 2 ⎟
∂u1 ⎟ ⎜ ∂u 3K m 1 Saffman 2
. (9)
⎟ ⎜ (u = Sgn − u ) ρ μ 1 2 1 W 2 = ⎜ ρ2 v2 ⎟ , ∂y 2πd ∂y ⎟ ⎜ ⎟ ⎜ ⎝ ρ2 E2 ⎠ The drag coefficient cD of particles as a function of their volume concentration is taken in the form used ρ2 Ec in [23]; for small volume concentrations, we take into ⎞ ⎛ ρ2 u 2 account the known dependence on the Mach number, ⎟ ⎜ which was often used in our previous research [24, 25]: ⎟ ⎜ ρ2 u22 + m2 p2 ⎟ ⎜ ⎟ ⎜ cD = cD1 , m2 0.08; cD = cD2 , m2 > 0.45; (4) F 2 = ⎜ ρ2 u2 v2 ⎟, ⎟ ⎜ ⎟ ⎜ cD = [(m2 − 0.08)cD2 + (0.45 − m2 )cD1 ]/0.37, ⎝ ρ2 u2 E2 + m2 u2 p2 ⎠ 0.08 < m2 0.45; ρ2 u2 Ec + ηm2 u2 p2
⎞ ⎛ 0.43 ρ2 v2 cD1 (Re, M) = 1 + exp − 4.67 ⎟ ⎜ M ⎟ ⎜ ρ2 u2 v2
⎟ ⎜ 4 24 ⎟ ⎜ +√ × 0.38 + , (10) G2 = ⎜ ρ2 v22 + m2 p2 ⎟; Re ⎟ ⎜ Re ⎟ ⎜ ⎝ ρ2 v2 E2 + m2 v2 p2 ⎠ 4 150m2 1.75 + cD2 (Re,m2 ) = , ρ2 v2 Ec + ηm2 v2 p2 3m1 (1 − m2 )Re ⎛ ⎞ √ ˜2 |u1 − u2 | ρ11 ρ11 d|u1 − u2 | Γ , M= ; Re = √ ⎜ ⎟
μ γ1 p ⎜ ⎟ ∂u2 m2 ∂v2 m2 ⎜ ⎟ Γ2 = ⎜ ηp1 (5) + − ⎟; 6m2 λ1 ∂x ∂y ⎝ ⎠ q= Nu(T1 − T2 ), d2 − I0 + η(f2x u2 + f2y v2 ) (11) Nu = 2 + 0.6Re1/2 Pr1/3 . ⎛ ⎞ 0 Here d is the particle diameter, γ is the ratio of specific ⎜ ⎟ ∂m2 ⎜ ⎟ heats, λ1 is the thermal conductivity of the gas, Re, Nu, + f2x ⎜ p1 ⎟ ⎜ ⎟ ∂x and Pr are the Reynolds, Nusselt, and Prandtl numbers, ˜ ⎜ ⎟ (6) Γ2 = ⎜ ⎟. μ is the gas viscosity, and T is the temperature. ⎜ p ∂m2 + f ⎟ ⎜ 1 ⎟ 2y The equations of state of the phases are ∂y ⎝ ⎠ q2 + f2x u2 + f2y v2 Here u is the velocity, p is the pressure, m is the volume concentration, ρ is the mean density, ρi = ρii mi , ρii is the inherent (true) density of the phases, f is the force of interphase interaction, q is the heat transfer between
p1 = ρ11 RT1 , m2 p2 = m2 p1 + pc , pc = G(m2 )ρ2 ec ;
(12)
G(m2 ) = αt [1 + 2(1 + ε)m2 g(m2 )]/2, g(m2 ) = [1 − (m2 /m∗ )4m∗ /3 ]−1 ;
(13)
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Khmel’ and Fedorov E1 = cv1 T1 + u21 /2, E2 = ec + cv2 T2 + u22 /2 + Q, Ec = ec + ηu22 /2,
(14)
where cv is the specific heat at constant volume and Q is the specific heat of the reactions. The particles are assumed to be incompressible (ρ22 = const). In Eq. (13), αt is the constant determined by the geometric k (k = 0.4 for spheres), roughness β, and restitution ε parameters:
2 a √ αt = , 1+ 3 b + a2 + b 2 (15)
2 1+β 1−k a = (1 − β 2 ) − 1 + ε2 , b = 2k . 1+k 1+k The dissipation of the energy of random motion is determined as 6 3/2 C0 ρ2 m2 g(m2 )(e3/2 − ec0 ), I0 = c πd2 (16) π 3/2 2 2 k + αr /αt C0 = αt 1 − ε + (1 − β ) . 2 1+k Here ec0 is the minimum value of the energy of random motion of particles (initial level of randomization). The domain of applicability of the hydrodynamic approach for the description of the motion of a granular medium consisting of colliding √ spherical particles is determined by the condition ε 0.57 − 0.43β [20]. In the simplest case of ideally smooth elastic spheres αt = 4/3 and their small concentrations, the discrete medium is similar to a monatomic gas with γ = 5/3. The dependences of the parameters η and C0 characterizing the effect of imperfection of collisions on the restitution ε and roughness β parameters were discussed in [20]. To solve Eqs. (1)–(14), we use a method previously applied for similar problems (e.g., in [24, 25]), which was described and tested in [26, 27]. The method is based on a TVD scheme for the gas and Jentry–Martin– Daly scheme for the particles. In the present work, the method is modified by introducing approximations of terms associated with random pressure and collision enp1 ∂m2 , ergy, as well as non-divergent terms of the form ∂x p1 ∂m2 p1 ∂u2 m2 p1 ∂v2 m2 , , and . ∂y ∂x ∂y
EFFECT OF THE SHOCK WAVE SLIDING ALONG THE PARTICLE LAYER ON THE FLOW STRUCTURE As applied to the particle dispersion processes, the mathematical model of a dense gas suspension with different pressures of the phases and the numerical method
of implementation of some initial-boundary-value problems for the model were preliminary tested by using the problem of shock wave interaction with a dense layer of dust. The initial conditions for this problem, which is of interest by itself, correspond to the experimental conditions [10]. The experiments [10] were performed in a rectangular chamber with a square cross section of the air-filled space (2 × 2 cm). Dust was placed into a cavity in such a way that the upper boundary of the dust layer coincided with the wall of the acceleration chamber. A mixture of starch with a minor fraction of lead oxide particles was used to obtain the internal structure of the layer by means of x-ray photography. The mean particle diameter was 10 μm. The experimental schlieren pattern [10] demonstrates the shape of the shock wave front and the pattern of dispersion of the upper layer of starch particles. X-ray patterns display a refracted surface of the main layer and the internal structure (densification wedge) behind the shock wave refracted in the layer. Owing to the high density of the layer and, correspondingly, small refraction angles, shock wave reflection from the wall inside the layer cannot be seen. Fan et al. [10] provided measured angles between the shock wave front and the layer surface, refraction angles of the layer surface and the shock wave in the layer, and some results calculated by the model of a two-phase granular medium [17] with allowance for the Saffman force, but with the Magnus force being ignored. The presented distributions of the particle density do not agree with the experimental patterns of dust lifting directly behind the shock wave. Nevertheless, for the above-mentioned angles, the numerical values as functions of the shock wave Mach number are consistent with the experimental data. Our calculations were performed for a monodisperse suspension of spherical particles 10 μm in diameter. The formulation of the initial-boundary-value problem corresponds to interaction of a plane shock wave propagating over a channel 3 cm wide with a rectangular semi-infinite cloud 1 cm high. At a sufficient distance from the cloud edge, the conditions of shock wave passage along the dust layer are consistent with the test conditions [10]. The initial mean density of the suspension was taken to be ρ20 = 520 kg/m3 in accordance with [10]; thus, the volume concentration of the suspension was 0.4; the thermodynamic parameters were those taken for starch particles. The shock wave Mach number was varied from 2 to 4. The following parameters of collision dynamics of particles were used: η = 1, C0 = 0.01, and ec0 = 0.00001 m2 /ms2 , which corresponds to collisions with small losses (of absolutely elastic, almost smooth particles) at the mean velocity of random motion approximately equal to 5 m/s. The
Effect of Collision Dynamics of Particles on the Processes of Shock Wave Dispersion
211
Fig. 2. Dispersion of the particle layer with allowance for the Magnus force (M0 = 2).
Angles of layer compression and shock wave refraction versus the Mach number α1 M0
Fig. 1. Sliding of the shock wave along the layer: (a) particle density field (M0 = 2); (b) numerical schlieren pattern (M0 = 4).
computations were performed on a rectangular uniform difference grid with a step Δx = Δy = 0.0001 m. Figure 1 shows the patterns of shock wave interaction with a dense dust layer (with the Magnus force being ignored): the shock wave slides along the layer, passes through the layer, and impinges onto the substrate. At small concentrations of particles, the shock wave refracted in the layer is reflected from the substrate and forms the Mach configuration [4]. As the particle concentration in the mixture is increased, this configuration transforms to the regular configuration [6, 28]. In a dense mixture with a sufficient depth of the layer, reflection may have a dispersion character because, as the shock wave enters a deep layer, it may transform to a dispersion compression wave [29, 30]. The angle of shock wave refraction in the layer α1 and the angle of refraction of the layer surface α2 measured in the experiments [10] are shown in the pattern of the particle density distribution (Fig. 1a). The numerical schlieren pattern (Fig. 1b) displays the shape of the shock wave front, which agrees well with the shock wave front in the experimental schlieren picture [10]. However, the shape of the dust layer surface in Fig. 1a does not completely coincide with the dust lifting pattern in the experimental photograph. This fact may indicate that particle collisions at moderate levels of randomization (ec0 0.00001 m2 /ms2 ) do not serve as the main factor of dust lifting in the considered problem.
α2
experiment [6]
calculation
experiment [6]
calculation
2
4.1
4.90
1.2
1.25
3
3.95
4.83
1.4
1.41
3.5
3.9
4.78
1.6
1.64
4
3.7
4.16
1.8
1.83
Taking into account the Magnus force makes it possible to reach good agreement between the calculated results and experimental observations of dust layer dispersion behind the shock wave. Figure 2 shows the particle density field for the Magnus coefficient KMagnus = 20 (at this value of KMagnus , good agreement between the calculated and experimental data on dust suspension cloud formation was reached in [12]). It is seen that the layer surface shape immediately after the shock wave transition differs from that in Fig. 1a. Thus, the Magnus force plays a key role in lifting of dust suspension particles directly behind the shock wave sliding along the layer. It should be noted that the shock wave structure inside the layer and the values of the angles α1 and α2 in the calculations with the Magnus force being ignored and at KMagnus = 20 coincide. Variation of the Saffman force coefficient KSaffman from zero to 160, i.e., in the range given in [8], shows that these forces do not exert any noticeable effect on dispersion of a dense dust layer, in contrast to the dynamics of lifting of individual particles [8]. The calculated angles α1 and α2 are compared in the table with the measured data [10]. The qualitative dependences on the shock wave Mach number are in good agreement; reasonable quantitative agreement is observed for the angle α1 , and good agreement is seen
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Fig. 3. Initial stage of interaction of a blast wave with a dense layer of the dust suspension. Gas pressure fields (M0 = 3 and Δt = 0.2 ms).
Fig. 4. Effect of the shape of the layer surface on its interaction with the shock wave: (a) smooth layer; (b) wavy layer; the pictures on the left and right are the numerical schlieren patterns and the particle density fields, respectively; M0 = 3.
for the angle α2 . A possible reason for the discrepancy of the calculated and experimental values of α1 is the fact that the calculations used the correct data for the mean density of the suspension, but the presence of a small fraction of heavy particles of the lead oxide was ignored: unfortunately, the fraction of lead oxide particles was not indicated in [10].
INTERACTION OF A BLAST WAVE WITH A DUST SUSPENSION LAYER We consider a problem of interaction of a blast wave (similar to that arising in a local explosion of methane in coal mines) with a dense layer of particles in the domain x 0, 0 y h(x). The source of the blast wave is lo-
Effect of Collision Dynamics of Particles on the Processes of Shock Wave Dispersion
213
Fig. 5. Effect of the Magnus force on dust layer dispersion behind the shock wave: ec0 = 0.00001 m2 /ms2 , m0 = 0.2, M0 = 3, and t = 1.6 ms; KMagnus = 1 (a), 20 (b), and 85 (c); the isolines marked by the number 5 show the results for the particle density of 5 kg/m3 .
cated at a certain height above the surface (see frame 1 in Fig. 3). In the two-dimensional formulation of the problem, the initial situation is a cylindrical shock wave with the axis of symmetry normal to the plane of the figure (the cylinder radius is 0.05 m, and the Mach number is varied). The dust surface can be either smooth or rough. The surface roughness is modeled by a sinusoidal dependence of the surface shape h = h0 + Δh cos(ωx); the parameters Δh and ω are varied. The problem is solved in the two-dimensional formulation. The computations are performed for spherical coal particles with a diameter d = 50 μm, and their volume concentration is varied from 0.1 to 0.4. The computational domain is 3 × 0.34 m, and the grid step is 0.001 m. A typical pattern of the initial stage of the process of shock wave propagation and its interaction with the dust layer is shown in Fig. 3 in the form of gas pressure fields at different time instants with a step of 0.2 ms. In the case considered here, a transverse wave is formed after shock wave reflection from the layer; the nonuniformity of the front of this transverse wave is caused by nonuniformity of the flow behind the front of the blast shock
wave sliding along the layer during its deceleration. The distribution of the particle density is similar to the pattern presented for a later time instant in Fig. 4a. Effect of the Saffman Force As in the previous problem, variation of KSaffman from 0 to 160 does not reveal any noticeable effect on the particle lifting pattern. Thus, the Saffman force makes practically no contribution to dispersion in the case of shock wave interaction with high-density dust layers, which differs this process from the dynamics of lifting of individual particles [8]. Effect of the Surface Shape The influence of layer surface waviness on interaction with the shock wave is illustrated in Fig. 4, which shows the process for M0 = 3, ρ20 = 240 kg/m3 , ec0 = 0.00001 m2 /ms2 , h0 = 2 cm, and Δh = 0 (Fig. 4a) and Δh = 0.2 cm (Fig. 4b) at the time instant of 2 ms. In this case, the Magnus force is ignored (KMagnus = 0). It is seen that the influence of the surface shape is man-
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Fig. 6. Isolines of the particle density with collisions being ignored (a) and taken into account (b): ec0 = 0.0001 m2 /ms2 , KMagnus = 0, M0 = 3, and t = 2 ms; the values of the particle density (in kg/m3 ) are marked by the numbers on the isolines.
ifested only in the patterns of shock wave interaction with the layer surface and does not affect the dispersion of particles from the layer. The patterns of the particle density in Fig. 4 are presented in a unified scale (which is nonuniform for visualization of the cloud at low densities). The maximum volume concentration of particles during layer compression reaches the value of 0.43. The Richtmyer–Meshkov instability may develop on the surface of the main compressed layer due to interaction with the shock wave penetrating into the layer (see, e.g., [31]). Effect of the Magnus Force A significant effect of the Magnus force in problems of dust lifting from dust layers under the action of shock waves was noted in [11–13]. The calculations by Eq. (8) with variation of KMagnus also testify to a significant effect of this force on the dispersion of dust layers. Figure 5 shows the particle density fields obtained for different values of KMagnus ; the isolines corresponding to the density of 5 kg/m3 are marked by the number 5. It should be noted that the dispersion pattern at KMagnus = 20 (Fig. 5b) provides the best agreement with the experimental photograph in [10]; at the same value, good agreement between the calculated and experimental results on formation of dust suspension clouds was obtained in [12].
Role of the System of Compression and Reflection Waves At high values of the Atwood number (about 100), the shock wave refracted in the layer propagates at a very small angle to the substrate and transforms to a dispersion compression wave inside the layer [30]. Therefore, the dust lifting mechanism associated with interaction of this wave with the substrate and the layer surface [4, 6] is possibly invalidated in dense layers by other factors: collisions of particles with each other and Magnus force effect. Effect of Collisions on Particle Lifting The calculations are performed for different values of parameters characterizing the intensity of random motion of particles. Figure 6 shows the results for a wavy layer (Δh = 0.2 cm) within the framework of the standard (collisionless) model (Fig. 6a) and with allowance for collisions with the initial random energy ec0 = 0.0001 m2 /ms2 corresponding to the mean velocity of oscillations equal to 10 m/s (Fig. 6b). In the presented variant, the Magnus force is ignored (KMagnus = 0). The calculations are performed for the particle density of 0.1, 5, and 20 kg/m3 . The location of the front for the case illustrated in Fig. 6 corresponds to the schlieren patterns in Fig. 4. It is seen that random motion ensures moderate “swelling” of the layer both in the initial state ahead of the shock wave front
Effect of Collision Dynamics of Particles on the Processes of Shock Wave Dispersion
Fig. 7. Typical distribution of the collision pressure (ec0 = 0.00001 m2 /ms2 , KMagnus m0 = 0.2, M0 = 3, and t = 2 ms).
215
=
0,
Fig. 8. Particle dispersion pattern for KMagnus = 20, m0 = 0.2, M0 = 3, t = 2 ms, and ec0 = 0.00001 m2 /ms2 . Density field (a) and density isolines (b).
and behind the front (Fig. 6b). Even in the absence of particle collisions, the motion of particles in the transverse direction (Fig. 6a) is more intense than that in the problem of sliding of a sustained shock wave along the layer (see Fig. 2 at KMagnus = 0). This happens because of a more complicated flow structure, in par-
ticular, owing to transverse inhomogeneity behind the front of the attenuated blast shock wave. A comparison of Figs. 6a and 6b shows that particle collisions promote the dispersion of the dust layer: the density isolines for 5 kg/m3 are weakly discernable in two models, but the isoline for 0.1 kg/m3 in the model
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Khmel’ and Fedorov
Fig. 9. Effect of the collision parameters on formation of dust clouds at t = 16 ms: ec0 = 0.00001 (a) and 0.0001 m2 /ms2 (b); KMagnus = 0, m0 = 0.1, and M0 = 5.
with particle collisions is appreciably higher; thus, because of particle collisions, the cloud boundary becomes significantly smeared. A typical distribution of the collision pressure is shown in Fig. 7. It can be noted that the maximum pressures in the discrete phase due to particle collisions are small as compared to the gas pressure, but significantly greater than the value of m2 p1 characterizing the contribution of the gas pressure (part of the buoyancy force) in the equations for the mean parameters of particles. The pattern of particle dispersion from the layer corresponding to KMagnus = 20 is shown in Fig. 8. It turns out that the effect of the collision dynamics for a significant initial level of random motion of particles is comparable with the Magnus effect (cf. Figs. 8b and 5b). The difference can be explained as follows: dust lifting due to particle collisions is characterized by a greater density gradient, i.e., smearing of the cloud boundary, while dust lifting induced by the Magnus force incorporates the entire mass of the upper layer. This is seen from the positions of the isolines for ρ2 = 0.1 kg/m3 and ρ2 = 5 kg/m3 in Figs. 5b and 8b. The influence of collision effects on the late stages of cloud formation is illustrated in Fig. 9, where the particle density fields are presented in a unified scale (calculation for KMagnus = 0). It is seen that an increase in the energy of random motion of particles by an
order of magnitude (i.e., an approximately three-fold increase in the velocity of oscillations) leads both to more intense “swelling” of the main layer (approximately by a factor of 2) and to an increase in the height of lifting of the sheet of the dispersed upper layer (by a factor of 1.5). Thus, the random motion and collisions of particles are important mechanisms of dispersion of dust suspension clouds behind shock waves. Development of the Kelvin–Helmholtz Instability The patterns of formation of dust clouds at the late stages (see Fig. 9) also show the development of the Kelvin–Helmholtz instability on the interface between the main mass of the layer shifted along the surface and the dust cloud formed from the upper layer of particles. The instability of this type was demonstrated, e.g., in the numerical calculations [6] within the framework of the gas-dynamic model for the problem of shock wave interaction with a layer of a denser (colder) gas. As in [6], “swelling” of the main mass of the layer during the development of the Kelvin–Helmholtz instability occurs with a considerable delay after the instant of the shock wave transition. This fact testifies that this mechanism of dust lifting develops within a certain time after passing of the shock wave; thus, it plays a secondary role as compared to other factors acting directly behind the shock wave. In [6], this is the formation of a
Effect of Collision Dynamics of Particles on the Processes of Shock Wave Dispersion wave system inside the layer at moderate values of the Atwood number; in our case of the dense layer, this is the Magnus force and the process of particle collisions. It should be noted that expansion of the main mass of the layer in which the Kelvin–Helmholtz instability develops is also more intense at a high level of random motion of particles.
CONCLUSIONS Problems of dust lifting from a bulk density layer due to passing of a plane shock wave and a blast wave are considered within the framework of a physicomathematical model of wave dynamics of a two-phase medium consisting of a gas and incompressible particles with due allowance for random motion and collisions of particles. Our previously developed numerical method for high concentrations of particles, which is based on schemes with operators providing monotonicity (Harten TVD and Jentry–Martin–Daly schemes) is modified for solving boundary-value problems of this model. The mathematical model is verified on the basis of the problem of passing of a plane shock wave along a dense layer. In particular, good agreement is obtained with experimental data on the shock wave shape, angle of shock wave reflection in the layer, angle of compression of the layer surface, and pattern of particle dispersion from the surface layer. An important role of the Magnus force in the process of particle lifting from the layer is demonstrated. In the problem of interaction of a blast shock wave with the layer, the Saffman force and the shock wave structure inside the layer (reflections) at high particle concentrations are found to exert a weak effect. The Magnus force is demonstrated to play a key role in the process of dust lifting directly behind the shock wave. At a significant initial level of oscillating characteristics of the suspension, the effect of particle collisions is comparable with the effect of the Magnus force. Collision effects are more pronounced in the patterns of formation of dust clouds at the late stages, as well as the development of the Kelvin–Helmholtz instability. Thus, the development of random motion and collision dynamics of particles is an additional important factor of dispersion of dense dust layers in shock wave processes. This work was supported by the Russian Foundation for Basic Research (Grant No. 16-08-00778) and by the Russian Science Foundation (Grant No. 16-19-00010).
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