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Combustion, Explosion, and Shock Waves, Vol. 44, No. 1, pp. 76–85, 2008

Numerical Study of Shock-Wave Diffraction in Variable-Section Channels in Gas Suspensions A. V. Fedorov,1 Yu. V. Kratova,1 and T. A. Khmel’1

UDC 532.529

Translated from Fizika Goreniya i Vzryva, Vol. 44, No. 1, pp. 85–95, January–February, 2008. Original article submitted December 29, 2006.

The problem of shock-wave transition past a backward-facing step in a gas suspension is solved. The calculation method is tested on a similar problem for a pure gas, and good agreement with available experimental and numerical results is reached. The effect of shock-wave intensity, mass load factor of particles in the mixture, and particle size on the flow structure in the gas suspension is determined. It is shown that the greatest difference between the flow pattern in a two-phase mixture and the corresponding flow in a pure gas is observed in the range of times when the characteristic sizes of the structures being formed are commensurable with the scale of the relaxation zones. Key words: diffraction of shock waves, two-phase flows, numerical simulation.

INTRODUCTION Diffraction of shock waves is one of the basic problems of gas dynamics and mechanics of multiphase media. Of particular interest is the shock-wave diffraction due to sudden expansion of a plane channel. Such a configuration is typical of channels in various engineering devices. The processes developed with time in the flow behind a backward-facing step are interaction of the shock wave with the vortex and the shock wave with the shear layer; there arise various types of reflection of the near-wall shock wave, a secondary shock wave, and pairs of shock waves induced by the vortex [1]. Some aspects of this phenomenon were studied both experimentally and by means of numerical simulations (a detailed review of papers dealing with this problem, as applied to gas mixtures, can be found in [2]). The problem of shock-wave (SW) diffraction on a rectangular configuration was used as a test in numerical simulations of unsteady compressible flows in a gas [3], where a comparative analysis of experimental data and results predicted by various numerical techniques was performed.

1

Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090; [email protected].

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The processes of SW diffraction in mixtures of a gas and solid particles are, obviously, more complicated than those in gas mixtures. The influence of velocity and temperature relaxation of two phases, whose characteristic duration is determined by the particle size, introduce an additional geometric scale. For this reason, the flow in the vicinity of the step is not self-similar, in contrast to the pure gas flow. From this viewpoint, an analysis of the flow behind a step with a shock wave passing in the gas suspension is of fundamental theoretical interest. Note that such issues have not been adequately studied, and there is little information available in publications. Wang et al. [4] solved the problem of propagation of a plane SW above a square cavity filled by a motionless gas suspension numerically. They found that the shock waves in the cavity become attenuated with increasing load factor of particles and transform to compression waves. The particle size exerts a significant effect on the flow character and the wave structure in the cavity. For a suspension of coarse particles (250 μm), the flow structure was demonstrated to approach that in gas mixtures (obviously, because the relaxation zones behind the SW in the case considered were greater than the size of the cavity, and the solution was determined by the frozen flow in the gas).

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Numerical Study of Shock-Wave Diffraction in Variable-Section Channels in Gas Suspensions A similar problem for a trapezoid cavity was considered in [5] within the framework of the approximation of single particles (without allowance for the effect of particles on the gas flow). It was shown that lifting of particles from the cavity walls is significantly affected by the SW intensity, initial position of the particle, and its size. The motion of particles in the upstream direction (saltation) was observed, which can lead to stable accumulation of particles near the compression corner behind the step. Kutushev and Shorokhova [6] studied the processes of combustion and detonation in gas suspensions in tubes with sudden expansion. Conditions of passing of the detonation wave through a discontinuity in the channel cross section with particles of a unitary fuel with a fixed particle size (30 μm) were studied numerically. The mass load factor of particles in the mixture was found to have a substantial effect on the critical ratio of tube diameters corresponding to detonation failure. Thus, the data of previous research show that the size of particles of the discrete phase and their mass fraction in the mixture exert a significant effect on the shock-wave processes in regions of complicated geometry. For this reason, an analysis of SW diffraction in the case of sudden expansion of the channel filled by a gas suspension is of both applied and theoretical interest. Wave structures arising due to diffraction of shock waves of moderate intensity in the case of a sudden change in the cross section of a plane channel are analyzed in the present paper on the basis of a physicomathematical model of motion of a two-phase medium within the framework of two-velocity two-temperature mechanics of heterogeneous media and numerical simulations of two-dimensional flows. The working media filling the channel are nonreacting (without ignition and combustion) mixtures of aluminum particles of different diameters (but of an identical diameter in each particular mixture) and oxygen with low and medium mass load factors of particles (lower than the stoichiometric value). The challenges of the present activities were: • to develop a numerical method for studying shockwave flows of gas suspensions in variable-section channels; • to simulate the process of shock-wave propagation in a gas suspension in a channel with sudden expansion numerically; • to study the influence of the mass load factor and size of particles on the wave structure of the flow in a gas suspension in the case of SW diffraction due to a sudden change in the channel cross section.

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Fig. 1. Flow pattern.

FORMULATION OF THE PROBLEM We consider a plane channel consisting of a narrow part and a wide part. The channel is filled by a homogeneous mixture of a gas and fine aluminum particles. The channel is assumed to be symmetric with respect to the x axis; hence, it is sufficient to consider only its upper half (Fig. 1). A steady plane SW propagates through the gas suspension from left to right over the narrow part of the channel. We consider the process of the SW transition from the narrow part of the channel to the wide part and subsequent propagation of the shock wave over the wide part of the channel. In the flow pattern (Fig. 1), L1 is the position of the SW front at the initial time, L2 is the length of the narrow part of the channel, L is the length of the computational domain, H1 is the height of the narrow part of the channel, and H2 is the height of the wide part of the channel. The shock-wave flow in nonreacting gas suspensions is described by the following system of equations following from the laws of conservation of mass, momentum, and energy of each phase: ∂ρi ui ∂ρi υi ∂ρi + + = 0, ∂t ∂x ∂y ∂ρi ui ∂ ρi u2i + (2 − i)p ∂ρi ui υi + + = (−1)i−1 (−fx ), ∂t ∂x ∂y ∂(ρi ui υi ) ∂ ρi υi2 + (2 − i)p ∂ρi υi + + (1) ∂t ∂x ∂y = (−1)i−1 (−fy ), ∂ρi Ei ∂ [ρi ui (Ei + (2 − i)p/ρ1 )] + ∂t ∂x ∂ [ρi υi (Ei + (2 − i)p/ρ1 )] + ∂y = (−1)i−1 (−q − fx u2 − fy υ2 ). The model is closed by equations of state with allowance for the fact that the volume concentration of particles is small p = ρ1 RT,

Ei = (u2i + υi2 )/2 + cυ,i Ti ,

(2)

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Fedorov, Kratova, and Khmel’

and by the laws of velocity and heat exchange between the phases: 3m2 ρ11 f¯ = cD |¯ u1 − u ¯2 |(¯ u1 − u ¯2 ), 4d

(3)

6m2 λ1 q= Nu(T − T2 ). d2 To take into account the dependence of the drag coefficient cD on the Reynolds and Mach number of the relative motion of particles, we use the formula derived and tested in [7] through comparisons with experimental data on the trajectories of particle motion under a shock-wave action: cD (Re, M12 ) 4 0.43 24 +√ = 1 + exp − 4.67 0.38 + , M12 Re Re (4) Nu = 2 + 0.6Re1/2 Pr 1/3 , M12

Re =

ρ11 d|u1 − u2 | , μ

√ |u1 − u2 | ρ11 = . √ γ1 p

In Eqs. (1)–(4), p is the pressure, ρi , ui , vi , Ei , and cv,i are the mean density, the streamwise and crossflow components of velocity, total energy per unit mass, and specific heat of the ith phase (i = 1, 2; the subscripts 1 and 2 refer to the gas and particles, respectively), m2 is the volume concentration of particles, T and T2 are the gas and particle temperature, ρ = ρ1 + ρ2 , d is the particle diameter, cD is the drag coefficient of particles, λ1 is the thermal conductivity of the gas, Re, Nu, and Pr are the Reynolds, Nusselt, and Prandtl numbers, respectively, μ is the gas viscosity, and γ1 is the ratio of specific heats of the gas. System (1)–(4) is solved under the initial conditions 0 < x < L1 , ϕl, (5) t = 0: ϕ = ϕ0 , L1 < x L, where ϕ = {ρ1 , ρ2 , ρ1 u1 , ρ2 u2 , ρ1 E1 , ρ2 E2 } is the vector of the solution, ϕl is the steady-state solution corresponding to a steady plane SW, and ϕ0 is the initial state ahead of the front. The initial values of parameters of the mixture were taken identical to those used in [8]: p0 = 1 atm, T0 = T20 = 300 K, cv,2 = 880 J/(kg · K), and cv,1 = 914 J/(kg · K). The particle size was varied within 1– 5 μm; the initial mean density of particles ρ2,0 was varied from zero (gas without particles) to 0.7 kg/m3 . The geometric parameters of the channel were H1 = 0.01 and 0.02 m, H2 = 2H1 , and L2 = 0.01 and 0.03 m.

CALCULATION METHODS The basic numerical method was the method successfully used in [9] for calculating two-dimensional unsteady detonation flows of gas suspensions of reacting particles within the framework of the two-velocity twotemperature approach of mechanics of heterogeneous media. A total variation diminishing (TVD) scheme [10] with a five-point template was used for the gas-phase equations. The Gentry–Martin–Daly scheme with upstream differences [11] was applied to solve equations that describe the solid-phase dynamics. The present numerical method has the following specific feature. For convenience of numerical implementation of the two-dimensional TVD scheme in the volume of a hybrid geometry, the computational domain was a plane channel of the maximum height. The computation was performed in the entire domain at each time step, after which the boundary conditions on the walls of the narrow and wide parts of the channel are determined anew in accordance with the no-slip conditions. The condition of a supporting piston for the SW in the gas suspension (value of the final equilibrium state) was set on the input (left) boundary. The computation was continued until the front reached the right boundary of the domain; therefore, the condition on the right boundary was an undisturbed flow of the gas suspension (initial state).

TESTING OF THE METHOD ON THE PROBLEM OF SW DIFFRACTION IN THE GAS As a test problem, we consider the diffraction of a plane SW with an intensity corresponding to M = 1.5 on a backward-facing step in a gas without solid particles (air). Let us note the typical features of the flow in the gas with the shock wave passing over the step. Figure 2 shows the flow pattern observed in experiments and obtained on Schlieren pictures [12]. When the SW leaves the narrow part of the channel and enters the wide part of the channel, rarefaction waves arise on the expansion corner, and the front of the incident SW becomes curved. The shape of the fan of rarefaction waves is related to the flow behind the incident SW. The flow whose scheme is shown in Fig. 2 is supersonic. In the case of low shock-wave Mach numbers typical of a subsonic flow behind the front, the characteristics of rarefaction waves propagate upstream of the step, and the shape of these characteristics becomes more rounded (Fig. 3).

Numerical Study of Shock-Wave Diffraction in Variable-Section Channels in Gas Suspensions

Fig. 2. Flow pattern in the case of shock-wave diffraction on a backward-facing step in gases [1].

The shear layer formed owing to flow separation from the surface of the narrow part of the channel coils into a vortex. A secondary shock is formed near the vortex, which allows the conditions behind the curved SW and the expanding flow to be matched. In addition, the secondary shock, like the vortex, allows matching the conditions behind the transient and curves shock waves. A curved contact surface is adjacent to the point of intersection of the incident plane shock and the outermost characteristic of the fan of rarefaction waves. This contact surface separates the gas that passed through the transient plane SW and the gas behind the diffracted wave. These features (contact surface and secondary shock) are most clearly expressed for rather high values of the Mach number (M > 2). The above-mentioned features of the flow were reproduced in the present numerical simulations whose results are plotted in Fig. 3c for a subsonic flow on the step after SW passing (SW Mach number M = 1.5). For comparison, Figs. 3a and 3b show the data from the review [3]: results computed by E. I. Lottati with the use of Godunov’s scheme in Fig. 3a and results computed by M. Watanabe (Harten–Yee TVD scheme) in Fig. 3b. A comparison of Fig. 3c with Figs. 3a and 3b, and also with Fig. 2 reveals reasonable agreement in terms of the flow structure on the expansion corner both with previous computations performed by different methods and with the flow pattern in Fig. 2. For a further study of similar flows of gas suspensions, it seems of interest to give some data on the influence of SW intensity on the flow pattern. The gas density contours for three Mach numbers are plotted in Fig. 4. It is seen that the primary effect of the increase in the Mach number is a qualitative change in

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the fan of rarefaction waves on the step. The flow on the step in Fig. 4a (M = 2) is subsonic and similar to the patterns in Fig. 3. As the Mach number increases, the flow on the step becomes supersonic (Figs. 4b and 4c); in this case, all characteristics of rarefaction waves emanate from the corner point (as in Fig. 2). With increasing SW intensity, the flow behind the step also changes. In particular, the region occupied by the fan of rarefaction waves increases and becomes more extended in the streamwise direction. The secondary shock and the contact surface are more expressed in Figs. 4b and 4c, which validates the growth of their intensity. The size of the vortex region also increases, and its shape becomes more extended. An analysis of the flow structure near the wall of the backward-facing step reveals the formation of the Mach stem at M = 3 and 4. This result is consistent with the data of experimental and numerical investigations [1, 12], where it was shown that the SW is directed normal to the wall at comparatively low Mach numbers (M < 2). Under moderate shock loads (M ≈ 3–4), the Mach reflection is developed near the wall, and the reflection become regular for strong shock waves (M > 4). This can be seen in Fig. 4c, where the Mach stem can be still seen, but its height is very small. Thus, the results of test computations for a gas without solid particles are in good agreement with available experimental and numerical data, which allows us to use the numerical technology tested to analyze the process of SW diffraction in a two-phase mixture.

SHOCK-WAVE DIFFRACTION IN A GAS SUSPENSION A distinctive feature of shock-wave flows in heterogeneous mixtures, as compared with inviscid gas flows, is also the fact that the internal SW structure is determined by the processes of thermal and velocity relaxation of the phases. The characteristic scales of the relaxation processes depend on the particle size; for the considered suspensions of identical aluminum particles in oxygen (in a particular suspension, the particle size could take a value between 1 and 5 μm), these scales are commensurable with the geometric scales of the problem (channel height). The presence of these relaxation processes can exert a significant effect both on the pattern of SW diffraction on the backward-facing step and on the conditions of further SW propagation in the channel. (Note that the relaxation processes exert a particularly profound effect on conditions of initiation, propagation, and failure of detonation waves in reacting gas suspensions.)

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Fig. 3. Computed patterns of shock-wave diffraction on a backward-facing step in air. Gas density contours (M = 1.5): the computations were performed by E. I. Lottati [3] (a), M. Watanabe [3] (b), and present authors (c).

The data for the flow formed when a shock wave moves over a square expansion corner in a gas suspension of aluminum particles 1 μm in diameter in oxygen with a fixed initial load factor of particles ρ2,0 = 0.69 kg/m3 were computed for a range of Mach number from 2 to 4 and are plotted in Figs. 5 and 6. Figure 5 shows the density fields of the gaseous and disperse phases and the vector field of gas velocities for M = 3,

i.e., for a supersonic flow around a step. Note, the wave pattern as a whole is similar to the flow with SW diffraction in a gas without solid particles (see Fig. 4b). Here we can also identify a vortex, a secondary shock, and a rarefaction wave, whose boundaries separate regions of different intensity of color in Fig. 5a. The presence of particles, however, has an essential effect on the shape and size of the characteristic structures of the flow.


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Fig. 4. Effect of SW intensity on the flow pattern in a gas without solid particles. Gas density contours (t = 16 µsec).

Formation of a Region with a Reduced Fraction of Particles. An analysis of the particle density contours (Fig. 5b) shows that a rarefaction zone with an extremely low load factor of particles (the mean particle density is smaller than 0.05 kg/m3 ) is formed behind the step. There seems to be the following reason. Immediately after the shock wave passes the step, the gas sharply changes the direction of its motion (Fig. 5c), and the particles by virtue of their inertia continue to move in the streamwise direction for some more time. Thus, a zone with no incoming particles from the flow region upstream of the step is formed behind the step. The particles that were initially located in this zone behind the step are entrained by the diffracted SW away from this region. Two phenomena occur here: the first one, caused by the leading diffracted SW, entrains the particles from the zone; the second phenomenon, caused by particle inertia, prevents their incoming to this zone. The vortex flow of the gas formed behind the step favors further separation of particles. Accumulation of particles occurs in the layer adjacent to the contact surface, which can be clearly seen in Fig. 5b. As the velocity of the discrete phase (and the gas phase as well) in the fan of rarefaction waves is substantially higher than that behind the diffracted SW (because of attenuation of the latter behind the step, Fig. 5c), the particles are decelerated in the region between the contact surface and the secondary shock (see also Fig. 2). The consequence of this fact is a significant increase in particle concentration (the mean density of particles reaches 5.36 kg/m3 ). This means that the mean density of particles increases almost by an order of magnitude, i.e., there arises a ρ-layer, which is usually formed behind the SW. It should be emphasized

that the layer in our case originated near the contact discontinuity. A similar result was obtained in [13], where the problem of propagation of decaying spherical shock waves generated by an explosion of the central charge was considered. The motion of fine particles in the gas flow field without allowance for the particle influence on the gas was considered. An analysis of particle trajectories showed that they form a ρ-layer ahead of the contact discontinuity separating the high-temperature layer of air from the products of detonation of the explosive substance. The Effect of SW Intensity. The effect of SW intensity on the pattern of SW diffraction on a backward-facing step in the mixture is illustrated in Fig. 6, which shows the gas density contours at an identical time for three different initial Mach numbers. Comparing this figure with Fig. 4, we can see that the characteristic features of the influence of the Mach number on the flow structure are also retained in the heterogeneous mixture. As the incident SW Mach number increases, the region occupied by the fan of rarefaction waves and the vortex zone become more extended. The pattern of reflection of the diffracted SW from the wall of the backward-facing step here is similar to the patterns of the flow in a gas without solid particles at the same Mach numbers. The presence of particles mainly affects the flow structure directly behind the front of the leading SW. The presence of relaxation processes alters the direction and shapes of the contours of the characteristic parameters. Thus, because of interaction of the fan of rarefaction waves and relaxation zone, the contours of the gas density turn in the direction of the central part of the channel and become closed on the centerline (see Fig. 6),

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Fig. 5. Shock-wave diffraction on a backward-facing step in a gas suspension (M = 3 and t = 16 µsec): (a) gas density field; (b) particle density field; (c) vector field of gas velocities.

whereas these contours reach the front of the leading SW in a gas without solid particles (see Fig. 4). The density contours between the front of the diffracted SW and the contact surface are strongly curved in Fig. 6, while they are close to straight lines in Fig. 4 (in a gas without solid particles), which is indicative of the linear character of the density behavior in the normal direction to the front of the diffracted SW). Effect of the Particle Size. The length of the zones of relaxation processes in gas suspensions with particles of an identical size is known to depend on the

particle diameter. The influence of the particle size on the structure of the diffracted SW in the mixture is illustrated in Fig. 7, which shows the results computed for M = 2, ρ2,0 = 0.69 kg/m3 , t = 40 μsec, and particle diameters d = 1, 3, and 5 μm. By comparing Figs. 7a and 7b, we can see that an increase in the particle size alters the slope of the secondary shock; the shape of the vortex region and the fan of rarefaction waves is also changed. For d = 3 and 5 μm, the characteristic scales of the flow structures in the zone behind the step at this time are significantly smaller than the character-


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Fig. 6. Effect of SW intensity on the flow pattern. Gas density contours (t = 1.6 · 10−5 sec).

Fig. 7. Effect of the particle size on the wave pattern. Gas-phase density contours.

istic scale of the relaxation zone, which can be estimated as the distance between the front of the incident SW and the outermost density contour at y = 0.035 m (Figs. 7b and 7c). For that reason, the slope of the secondary shock in Figs. 7b and 7c is practically identical. For the maximum particle diameter considered (d = 5 μm), the relaxation zone is so large that the flow behind the step is actually frozen, i.e., it approaches the flow formed in a gas without solid particles. As the characteristic sizes of

the flow structures behind the step increase with time, at a certain moment they become commensurable with the characteristic scales of the relaxation zones for particles 3 and 5 μm in diameter; then the influence of the relaxation processes becomes also essential for the particles mentioned. For fine particles (d = 1 μm), vice versa, the scales of the relaxation processes are much smaller than the characteristic scales of the structures, and the flow approaches the equilibrium state. These

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Fig. 8. Effect of the particle size on the gas density distribution (y = 0.02 m and M = 2).

Fig. 9. Effect of the load factor of particles in the gas suspension on the flow pattern (M = 3 and t = 16 µsec).

results agree with conclusions obtained by analyzing the data calculated in [14], where the processes of SW interaction with a dusty cloud on the bottom of a plane channel were considered. The results also show that the pressure profile with decreasing particle diameter approaches the corresponding distribution obtained by the equilibrium model. Figure 8 shows the distribution of the gas-phase density along the channel at a step height y = 0.02 m at the time of 40 μsec (the step is located at x = 0.03 m).

It is seen that the density profiles are nonmonotonic and have both a point of a local maximum (which is related to the influence of the relaxation processes on the flow between the leading and the secondary shocks) and by a point of a local minimum (with a segment of the rarefaction wave adjacent to it). Note that the peak value of pressure at the point of the local maximum is significantly higher for fine particles. Here the influence of the relaxation processes for coarse particles is much less pronounced owing to a greater relaxation zone.

Numerical Study of Shock-Wave Diffraction in Variable-Section Channels in Gas Suspensions The Effect of the Load Factor of Particles. The effect of the load factor of particles on the diffraction pattern is illustrated in Fig. 9. If the content of particles in the mixture is small (ρ2,0 = 0.027 kg/m3 , Fig. 9a), the pattern almost coincides with that observed in a pure gas. Comparing the location of the front of the diffracted SW in Figs. 9b and 9c with that in Fig. 9a, we can see SW deceleration with increasing mass load factor. Note that the equilibrium velocity of sound ahead of the front decreases together with a decrease in SW velocity; thus, the SW Mach number remains almost unchanged. It is also seen in Figs. 9b and 9c that effects associated with the influence of the relaxation regions start manifesting if the concentration of particles increases by an order of magnitude and higher: curving of rarefaction waves and changes in the character of the flow between the front of the diffracted SW and contact discontinuity and the flow in the vortex zone. Thus, the effect of particles on the flow of gas suspensions becomes fairly significant in shock-wave processes in channels of complex geometry as soon as the relative mass concentration of particles reaches ξ2 = ρ20 /(ρ20 + ρ10 ) ≈ 0.1.

CONCLUSIONS • The problem of propagation of a shock wave in a gas suspension in a plane channel with a discontinuity in the cross section is studied numerically within the framework of a physicomathematical model of mechanics of heterogeneous media in the two-velocity twotemperature approach. • The flow structure as a whole in a mixture with shock-wave diffraction on the cross-sectional discontinuity is found to agree qualitatively with a similar flow in a pure gas. The presence of particles, however, affects the shape and size of the flow structures being formed. • The effect of the mass load factor of particles for their mass concentrations of ≈0.1 and higher is manifested in the shape of the fan of rarefaction waves, in the flow between the front of the diffracted shock wave and contact surface, and in the flow in the vortex zone. • The influence of the particle size on the diffraction pattern is most noticeably expressed in the time interval, where the characteristic scales of the structures are commensurable with the scales of the relaxation zones. This work was supported by the Russian Foundation for Basic Research (Grant No. 06-01-00299).

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