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Combustion, Explosion, and Shock Waves, Vol. 46, No. 5, pp. 578–588, 2010

Numerical Simulation of Shock Wave Propagation in a Mixture of a Gas and Solid Particles A. V. Fedorov1 and I. A. Fedorchenko1

UDC 532.529

Translated from Fizika Goreniya i Vzryva, Vol. 46, No. 5, pp. 97–107, September–October, 2010. Original article submitted September 20, 2009; revision submitted February 24, 2010.

A numerical method based on the cubic interpolated polynomial (CIP) approach is applied for simulation of two-velocity two-temperature two-phase flow dynamics. Validation of the results is provided by numerical tests. A problem of shock wave propagation in a mixture of a viscous heat-conducting gas and solid particles of essential volume fractions is investigated as an application case. The influence of the particle size and drag coefficient formulation on the flow pattern, in particular, on the temperature behavior within the relaxation zone is revealed. A comparison with experimental dependences of the parameters behind the shock front on the Mach number is performed. Key words: heterogeneous media, shock waves, mathematical modeling.

INTRODUCTION The need in prediction of the gas/ solid particle mixture behavior under the action of shock wave loading arises due to development of industrial explosion prevention methods and means of decreasing its destructive consequences, as well as in connection with design of solid-propellant engines of aircraft, along with some other problems of applied gasdynamics. To take into account nonequilibrium phenomena is important because the dynamic processes differ essentially as the shock wave propagates through the phases, for instance, within the areas of dynamic, thermal, and chemical relaxation (processes of ignition and combustion of the solid species) [1].

PHYSICOMATHEMATICAL MODEL The investigations presented in the paper were carried out within the frame of the heterogeneous medium mechanics approach based on multispeed continuum theory postulates. It is known that this theory allows macroscopic processes that occur in heterogeneous media to be described by analogy with those in homoge1

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

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neous flows by considering several (depending on the number of the phases) interpenetrating and interacting species [2]. The mathematical model of a mixture of a viscous heat-conducting gas and solid particles with the two-velocity two-temperature approach implemented includes conservation laws of mass, momentum, and energy for each phase with taking into account external forces, interphase interaction, and heat exchange: ∂ρ1 ∂ρ1 u1 ∂ρ2 ∂ρ2 u2 + = 0, + = 0, ∂t ∂x ∂t ∂x (1) 2 ∂u1 ∂p ∂u1 ∂ u1 + u1 = −m1 − f12 + μ ρ1 , ∂t ∂x ∂x ∂x2 ∂u2 ∂p ∂u2 + u2 = −m2 + f12 , ρ2 ∂t ∂x ∂x ∂e1 m1 p d1 ρ11 ∂e1 + u1 = ρ1 ∂t ∂x ρ11 dt 2 ∂u1 ∂ 2 T1 − f12 (u2 − u1 ) − q12 + λ , + μ ∂x ∂x2 ∂e2 ∂e2 + u2 = f12 (u2 − u1 ) + q12 , ρ2 ∂t ∂x p = ρ11 RT1 . Here ρi = ρii mi is the effective density of the ith phase, ρii is the physical density of the phase, ui , Ti , and ei are

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Numerical Simulation of Shock Wave Propagation in a Mixture of a Gas and Solid Particles the velocity, temperature, and internal energy of the phases, p is the mixture pressure, the subscripts i = 1 and 2 refer to the gas phase and the solid (disperse) phase, R is the air gas constant, f12 is the interphase 3 m2 λNu interaction force, q12 = (T1 − T2 ) is the heat 2 rp2 flux between the phases, μ is the gas viscosity with the Einstein correction, which takes into account the presence of particles in the flow, λ is the thermal conductivity of the gas, and rp is the particle radius. The volume fraction mi is used when multiphase media are considered, because each species of the mixture occupies only a certain part of the total volume. The local equilibrium hypothesis applied within each phase allows using thermodynamic functions and their correlations of equilibrium thermodynamics. Therefore, to have the system of the governing equations closed, the equation of state of the pure gas is implemented. We assume here that the solid phase is incompressible, i.e., ρ22 = const. The following three relations are applied to determine the drag coefficient CD needed for the interaction force assessment: CD = 27Re−0.84 , (2) 0.43 CD = 1 + exp − 4.67 M12 24 4 × 0.38 + +√ , (3) Re Re 24 (1 + 0.058Re0.9345 ). CD = (4) Re These equations are taken from the papers [3], [4], and [5], respectively. Equation (3) is a reduced form of the ρ11 |u2 −u1 |Dp Henderson equation [6]. Here, Re = μ is the Reynolds number, Dp is the particle diameter, |u2 −u1 | , a1 = ργp is the speed of sound in the M12 = 11 a1 gas, and γ is the ratio of specific heats of the gas.

NUMERICAL APPROACH A solution of a boundary-value problem of the wave process taking place in the gas–particle mixture based on the model described above was obtained by the cubic interpolated polynomial (CIP) scheme [7]. While being developed, the scheme was demanded to give an accurate prediction of propagation and interaction processes of contact discontinuities and shock waves. For this purpose, the profiles of the flow parameters inside each cell are interpolated by a cubic polynomial. Non-advective terms are approximated by the central finite-difference

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schemes. The discrepancy of the method is that it is nonconservative, which imposes some limitations on its application. Nevertheless, a check of stability and convergence of numerical results along with testing proved the reliability of the data obtained. The scheme details and results of solving a test Riemann problem are presented in [8].

PHYSICAL STATEMENT OF THE PROBLEM Let us consider a one-dimensional space filled with an air–solid particle mixture over which a strong discontinuity propagates. The flow in the shock wave (SW) propagating through such a mixture was described in many papers (see, e.g., [1, 9–12]). We assume that a quiescent mixture characterized by undisturbed flow parameters (ρ10 , ρ20 , u10 = u20 = 0, e10 , e20 , and p0 ) is located in front of the SW. The gas parameters at the frozen SW front change in a jump, and the parameters uf 1 , ρf 1 , ef 1 , and pf behind the front are determined by the given shock wave speed D with the aid of the Rankine–Hugoniot jump conditions for the gas, because the volume fraction of the particles is considered to be finite but low. The particle parameters at the front are believed to be constant, i.e., frozen, and are determined as follows: uf 2 = u20 , ρf 2 = ρ20 , and ef 2 = e20 . We need to determine the gas/particle mixture parameters within the propagating SW front.

RESULTS AND ANALYSIS Physicomathematical Model of Heterogeneous Medium Mechanics. Frozen Flow To start with, following [11], let us estimate a possibility of using the frozen SW conditions for determining the gas parameters. Figure 1 shows the computed dependence of the frozen gas pressure on the SW Mach number, which is obtained on the basis of the SW conditions for the pure gas and coincides with the experimental data [11] (points). The pressure value is normalized by the initial pressure in front of the SW. The particle material in both the experiments and computations is white soot with the physical density of 2650 kg/m3 . The coefficient η is the ratio of the particle mass flow ρ2 u 2 to the gas mass flow: η = . This parameter afρ1 u 1 fects the frozen flow parameters. We should note good agreement of the computed and experimental data. It confirms the validity of using the frozen parameters for finding the gas-phase characteristics [2].

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Fig. 1. Frozen pressure versus the SW Mach number. Comparison of the computed results (curves) and experiment [11] (points).

Equilibrium Flow To check the computational results obtained by the model, it is necessary to determine the equilibrium parameters, such as ue1 , ue2 , ρe1 , ρe2 , ee1 , ee2 , and pe . They are determined from the SW relations with the velocities and temperatures of the phases being assumed equal. Figure 2 shows the dependences of the equilibrium pressure behind the SW on the Mach number for different values of η. The curves correspond to the present calculations, and the points stand for the experimental data [11]. The computed data obtained in [11] are fully identical to our results and, therefore, are not shown in the figure.

Unsteady Approach. Shock Wave Structure in the Mixture Now we come to the SW structure problem investigation within the frame of the unsteady approach with the aid of the CIP method. In the first place, it is necessary to compare the unsteady and steady computational results. The steady flow parameters in the relaxation zone behind the frozen SW front were determined by a system of algebraic and differential equations derived from the initial system (1) within frame of the progressing wave approach: ρ1 u1 = c1 , ρ2 u2 = c2 , c1 u1 + c2 u2 + p = c3 , p u2 u2 + c2 e 2 + 2 = c 4 , c1 e 1 + 1 + 2 ρ1 2

Fig. 2. Equilibrium pressure versus the SW Mach number. Comparison of the computed results (curves) and experiment [11] (points).

p = ρ11 RT1 ,

e1 = cv1 T1 ,

e2 = cv2 T2 ,

(5)

du2 dp = −m2 + f12 , dx dx dT2 q1 = u2 . dx cv2 ρ2 For system (5) we pose the Cauchy problem for the solution vector Φ(ρi , ui , p, . . .): c2

Φ(ρi , ui , p, . . .) = Φf

at x = x0 + 0,

Φ(ρi , ui , p, . . .) → Φe

at x → ∞.

(6)

It means that the frozen SW conditions take place at x = x0 + 0 and the conditions of approaching the final equilibrium state are realized at x → ∞. Kazakov et al. [9] considered a possibility of solving this problem for a viscous isothermal gas and gave some numerical results obtained from steady computations. Below, the problem is solved numerically in the general case of variable temperatures of the phases and for a viscous heatconducting gas phase in the one-dimensional unsteady formulation. The parameter distributions along the wave obtained with the steady approach were used as initial data for solving the unsteady Cauchy problem. To test the method, a Riemann problem with equilibrium and frozen mixture parameters at the discontinuity was solved as well. Some results of the calculations are presented in Figs. 3 and 4. It is known that the SW in a mixture of an inviscid heat-non-conducting gas and particles consists of the frozen front accompanied by the velocity

Numerical Simulation of Shock Wave Propagation in a Mixture of a Gas and Solid Particles

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Fig. 3. Density (a), pressure (b), velocity (c), and energy (d) distributions of the phases for the particle radius equal to 20 µm, SW Mach number of 1.6, and particle volume fraction m20 = 0.001. The computations are performed with the drag coefficient equation (2).

and temperature relaxation zone till the final equilibrium state is attained. The wave structure obtained within the frame of the unsteady approach is shown in Figs. 3 and 4 as distributions of pressure, velocity, specific density, and internal energy for each phase. The density is normalized to the initial density of the pure gas ρ11,0 , the velocity to the gas-phase sound speed a1,0 , the pressure to ρ11,0 a21,0 , the energy to a21,0 , and the temperature to cv1 a21,0 , where cv1 is the gas-phase heat capacity at constant volume. These parameters are applied through the paper if it is not mentioned otherwise. It can be seen that the frozen SW in the flow is smeared due to viscosity and heat conduction, though not that strongly. This is conditioned by the low values of the dissipative coefficients. Moreover, it turned out that the SW front parameters and those behind the smeared shock in the relaxation zone grow due to the nonequilib-

rium processes of velocity and temperature relaxation. It is also shown that the numerical CIP method used for the computations conserves the SW structure with evolution in time. Thus, the SW propagation stability to infinitesimal disturbances is shown, as the steady solution used does carry on the computational errors, i.e., it is disturbed. It should be noted that the numerical method reproduces the smeared leading front of the SW well enough. Influence of the Drag Coefficient Formulation It is interesting to investigate the drag coefficient influence on the parameter distributions along the shock wave. To do this, we performed computations using relations (2), (3), and (4) (see Figs. 3, 4, and 5, respectively).

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Fig. 4. Distributions of velocity (a) and temperature (b) for the particle radius of 20 µm, SW Mach number of 1.6, and particle volume fraction m2 = 0.001. The computations are performed with the drag coefficient equation (3).

Fig. 5. Distributions of velocity (a) and temperature (b) of the phases for the particle radius of 20 µm, SW Mach number of 1.6, and particle volume fraction m20 = 0.001. The computations are performed with the drag coefficient equation (4).

Applying the correlation formulas (3) and (4) decreases the relaxation zone size, and the gas temperature distribution acquires a qualitatively different form. Thus, with the coefficient determined by Eqs. (3) and (4), the gas temperature first increases and then decreases down to the equilibrium value. With computations by Eq. (2), moderate cooling of the gas is first observed, and then it is heated up to the equilibrium temperature. This behavior of the gas-phase temperature can be explained by analyzing the heat balance equation for

the gas, which takes the following form: ∂T1 f12 q12 p dρ11 = − (u1 − u2 ) + . ∂x cv1 ρ1 u1 cv1 ρ211 dx cv1 ρ1 u1

(7)

It is known that the interaction force f12 depends on the drag coefficient CD as f12 =

ρ11 m2 3 CD (u1 − u2 )|u1 − u2 |. 8 rp

As the gas velocity in the relaxation zone is higher than the solid-phase velocity due to the inertia of the particles, the contribution of the first right-hand-side term


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Fig. 6. Contributions of the 1st, 2nd, and 3rd right-hand-side terms of Eq. (7) with the drag coefficient calculated by Eqs. (2) (a) and (3) (b).

Fig. 7. Gas temperature distribution versus the particle radius with the drag coefficient calculated by Eqs. (2) (a) and (3) (b).

of Eq. (7) is always positive, which is explained by the friction forces between the gas and particles. The second term is associated with the pressure forces, and this term also takes positive values because the gas is compressed by the transient SW, i.e., the gas density increases. The third term is responsible for the heat exchange between the gas and solid phases. As the particles are heated in the relaxation zone up to the gas temperature, this term in the heat balance equation has a negative sign. Depending on the choice of the equation for the drag coefficient, the first right-hand-side term of Eq. (7) responsible for the friction-induced heat release can take different values. With using Eq. (2), the temperature

gradient in the area right behind the discontinuity may become negative, because the third term contribution due to the heat transfer to be spent on particle heating prevails. As the value of u1 − u2 decreases, i.e., as the velocities and temperatures of the phases approach the equilibrium, the summed value of the first and second right-hand-side terms of Eq. (7) becomes larger, while the heat transfer from the gas toward the particles decreases. As a result, the gas temperature grows. The graphical representation of the contributions of all terms is given in Fig. 6a. When Eqs. (3) and (4) are used for determining the drag coefficient, the correlation of the right-handside terms of Eq. (7) is such that the gas temperature

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Fig. 8. Density (a) and velocity (b) distributions of the gas and the disperse phase computed in [12].

growth is observed first due to the heat release caused by velocity relaxation and the pressure force work, and then the heat transfer toward the particles dominates as the velocities relax. The evolution of the process is illustrated in Fig. 6b. This behavior is explained by the difference in the assumptions used to derive the empirical relations for the particle drag coefficient. Let us note the qualitative agreement of the temperature behavior with the use of the Henderson formulation [6] and the equation derived in [5] where unsteadiness of the flow around the particles is taken into account, in contrast to [3, 4]. The ratio of the heat release due to the friction forces and the heat transferred to particles is determined by a parameter [13] α = τT /τu , where τT and τu are the temperature and velocity relaxation times, which depend on the Reynolds number, Mach number, etc. With the particle size increasing, the maximum value of α grows. Figure 7a shows the gas temperature distribution for different particle radii with the drag coefficient CD calculated by Eq. (2). It can be seen that the qualitative behavior of the parameter in the relaxation zone does not depend on the particle radius; the gas temperature decreases right behind the leading SW front and then gradually increases up to the equilibrium value. The relaxation zone length grows naturally with the increase in the particle radius.

Fedorov and Fedorchenko As opposed to [13], where the influence of the relaxation time ratio was investigated for a steady flow with frozen velocity and temperature relaxation times, the influence of the “unsteady” drag coefficient [5] is also analyzed in the present paper. It depends on flow parameters in the shock wave. Figure 7b shows the gas temperature distribution for different particle sizes with the drag coefficient calculated by Eq. (3) [the temperature behavior with the drag coefficient calculated by Eq. (4) is similar to this one]. It turned out that the temperature behavior is independent of the particle size. Both for fine and coarse particles, an increase in the gas temperature is observed up to a value greater than the equilibrium one, with a subsequent decrease down to the final equilibrium state. Thus, the distribution of this quantity always has a local maximum, regardless of the particle size and the ratio α. As is shown in Fig. 7, the parameters at the smeared frozen SW front keep constant because they do not depend on the relaxation time. At the same time, a change in the radius leads to quantitative differences in the relaxation zone, in particular, in the temperature distribution.

Two Riemann Problems in the Gas–Particle Mixture The next step is to study the SW structure formed in the Riemann problem for the gas–particle mixture. The physical statement is as follows (see Table 1). At the initial time instant, region 2 in a one-dimensional channel on the left of the location x = x0 is filled with a pure gas at a high pressure; on the right of this point, region 1 contains the gas–particle mixture at atmospheric pressure. At t > 0, after the gas–mixture diaphragm rupture, a shock wave travels to the right, and a rarefaction wave (RW) goes to the left, a contact discontinuity (CD) in the gas follows the shock wave, and afterwards there appears a combined discontinuity (CombD), which divides the pure gas and the mixture. The initial data and the x–t diagram of the problem are presented in Table 1. The parameters of the computations were normalized by the corresponding values on the right of the initial discontinuity. Saito et al. [12] proposed two approaches to solving the equations describing the disperse phase behavior. These approaches were based on the finite volume method and differed by the way of data interpolation at the interface. The results of the simulation [12] by one of the methods, which appeared to be less dissipative, are shown in Fig. 8.


Fig. 9. Density (a), velocity (b), pressure (c), and temperature (d) distributions of the phases obtained from solving the Riemann problem.

Fig. 10. Solution convergence on a sequence of refined grids. Density and pressure distributions obtained from solving the Riemann problem.

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Fedorov and Fedorchenko TABLE 1

The computed results of the Riemann problem by the CIP method are presented in Fig. 9. The initial data are drawn by the solid curves (thin for the gas phase and thick for the solid phase). The calculations at the time t = 2 are shown by the dashed curves (thin for the gas phase and thick for the solid phase). The particle density is 2500 kg/m3 ; the ratio of specific heats is assumed to be equal to 1. The grid step is h = 0.1, and the time step to the space step ratio is 0.2. The wave structure formed after the diaphragm rupture can be seen in Fig. 9. A steady SW with a smeared frozen front propagates to the right (see Fig. 9c). It is totally identical to that obtained within the frame of the steady approach. It means that stable steady SW propagation is again reached for this kind of initiation. Figure 9a shows the phase densities and also some other flow features, in particular, the gas contact discontinuity location. At the time instant shown, it is at the point x = 6.8. The combined discontinuity is visible from the disperse phase density distribution; it is located at x = 5.8. At the initial position of the solid-phase velocity discontinuity, a non-physical fluctuation is observed (similar to [12]), possibly induced by the numerical scheme (see Fig. 8b). Note that this fluctuation does not affect the distributions of other parameters. The data presented are obtained with viscosity and thermal conductivity of the gas phase taken into account. As opposed to inviscid computations, some smearing of the shock wave, a contact discontinuity, and a rarefaction wave can be seen in this case. In general, the influence of viscosity on the flow pattern is not significant; for this reason, these numerical results are not shown here. The CIP method convergence is demonstrated by considering this problem. The densities of the phases and the pressure distributions are shown in Fig. 10 for different grid steps. The time step to space step ratio is 0.2. It is seen that the solution converges in the vicinity of discontinuities as the number of grid cells is increased.

Region 1

Region 2

p=1

p = 10

ρ1 = 1

ρ1 = 1

ρ2 = 1

ρ2 = 10−6

u1 = 0

u1 = 0

u2 = 0

u2 = 0

TABLE 2 Region 1

Region 2

p=1

p = 3.46

ρ1 = 1

ρ1 = 3.46

ρ2 = 1

ρ2 = 10−6

u1 = 0

u1 = 0

u2 = 0

u2 = 0

With h = 0.025, the flow pattern possesses an accurate structure of contact and shock discontinuities, as well as of rarefaction waves. The second possible SW type in heterogeneous media mechanics is dispersed shock waves. The initial data defining the flow structure of such a totally dispersed SW are given in Table 2. The physical pattern of the flow realized is similar to the Riemann problem described above (see Table 1). The difference is only in the SW front, which has a continuous profile in the case of the dispersed wave because there is no frozen part of the SW where the values change rapidly. The results of the calculations for this problem are shown in Fig. 11. After the diaphragm rupture, a shock wave with continuous profiles of all parameters propagates, as is shown in Fig. 11. At the time presented in the figure, the contact and combined discontinuity locations coincide (Fig. 11b). A comparison of the numerical data obtained in the current paper and in [12] shows quantitative agreement of the parameter distributions, except for the solidphase velocity fluctuation in the vicinity of x ≈ 88, which disappears with time, as is seen in Fig. 11.


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Fig. 11. Density (a), pressure (b), velocity (c), and temperature (d) distributions of the phases in the problem of disperse shock wave propagation at two time instants.

CONCLUSIONS • The shock wave structure in a mixture of a viscous heat-conducting gas and solid particles is reproduced on the basis of the CIP method applied for simulation of unsteady two–velocity two–temperature heterogeneous flows, and stability of the flow structure to infinitesimal disturbances is shown. • The validity of the conditions on the frozen and equilibrium shock waves is observed, based on comparisons with experimental data. • A qualitative difference of temperature distributions within the relaxation zone behind the leading shock front is demonstrated, depending on the drag coefficient formulation. The work was supported by the Analytical Departmental Targeted Program “Development of the Sci-

entific Potential of the Higher School (2009)” (Grant No. 2.1.1/4674).

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