Combustion, Explosion, and Shock Waves, Vol. 41, No. 3, pp. 336–345, 2005
Computation of Dust Lifting behind a Shock Wave Sliding along the Layer. Verification of the Model A. V. Fedorov1 and I. A. Fedorchenko1
UDC 532.529
Translated from Fizika Goreniya i Vzryva, Vol. 41, No. 3, pp. 110–120, May–June, 2005. Original article submitted April 13, 2004.
The problem of dust lifting behind a shock wave is solved within the framework of the equilibrium model of mechanics of heterogeneous media. Verification of the model proposed is performed. It is shown that different flow patterns are formed in layers with different shapes of the edge and constant- or variable-amplitude shock waves. Allowance for turbulence of the mixture leads to origination of a high-velocity nearwall trickle at the edge of the layer, and the particles are lifted to a greater height. Key words: mixtures of gases and solid particles, shock waves, mixing.
INTRODUCTION The problem of dust lifting from an unsteady surface layer under the action of an initiating shock wave is of considerable interest in theory and in practice. The practical significance is caused, e.g., by the fact that this process is the first stage of development of “layer detonation,” whereas the theoretical importance is associated with the possibility of obtaining new information about the mechanism of mixing at the contact boundary between a heterogeneous medium (dust layer) and air. Some papers dealing with physical and mathematical aspects of this phenomenon, which have been investigated since early 1960s, are reviewed in [1]. The present work continues the research of [2–9] and deals with mathematical simulation of the problem mentioned above within the framework of mechanics of an equilibrium heterogeneous mixture with allowance for various physical phenomena, including turbulence of the continuous phase. Another objective of this work was verification of the mathematical model and computation method by means of qualitative comparisons with available experimental data and identification of the influence of the turbulent component on the wave pattern of the flow. 1
Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090;
[email protected].
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PHYSICOMATHEMATICAL FORMULATION OF THE PROBLEM. GOVERNING EQUATIONS Figure 1 shows the flow pattern in the considered two-dimensional unsteady problem of mechanics of heterogeneous media on lifting of a dust layer under the action of a shock wave (SW). The SW propagates from right to left over the gas with the parameters u0 = −D, ρ0 , T0 , and p0 . The Mach number of the initiating SW is Ms = D/c0 (c0 is the velocity of sound ahead of the SW front and D is the SW velocity). There is a layer of particles of thickness h on the wall ahead of the SW. The density of the layer is characterized by the volume concentration of the disperse phase in the mixture (m2 ). A flow with the parameters ρ2 , u2 −D, T2 , and p2 is formed in the pure gas behind the SW. The parameters ρ2 , T2 , and u2 at the initial time are determined by
Fig. 1. Flow pattern.
c 2005 Springer Science + Business Media, Inc. 0010-5082/05/4103-0336
Computation of Dust Lifting behind a Shock Wave Sliding along the Layer relations on the normal shock. The gas upstream of the SW has the parameters ρ0 = 1.177 kg/m3 and T0 = 288 K. The equations that describe the flow of the mixture of the gas and fine solid particles with allowance for turbulence of the carrier phase have the form ∂G ∂Q ∂F + + = 0, ∂t ∂x ∂y where
ρ ρu Q= ρv ρ(E + k) is the vector of mass variables, F and G are the vectors of conservative fluxes in the x and y directions, respectively. Here ρ is the density, u and v are the velocity components in the Cartesian coordinate system, E is the total energy, k is the turbulent kinetic energy, and p is the pressure in the mixture. Within the framework of the Boussinesq hypothesis, the Reynolds stresses are 4 ∂u 2 ∂v τxx = µt − , 3 ∂x 3 ∂y τxy = µt
∂u ∂y
∂v , ∂x
2 ∂u . 3 ∂y 3 ∂x In these equations, µt is the turbulent viscosity determined by the following dependence in Wilcox’s twoparameter k–ω model of turbulence [10]: τyy = µt
4 ∂v
+ −
k ω (ω is the specific dissipation rate of turbulent kinetic energy). The turbulent parameters k and ω are determined by solving the differential equations ∂αx ∂αy ∂ρk ∂ρuk ∂ρvk + + = + + Hk , ∂t ∂x ∂y ∂x ∂y µt = Cµ ρ
∂βx ∂βy ∂ρω ∂ρuω ∂ρvω + + = + + Hω , ∂t ∂x ∂y ∂x ∂y where αx , αy , βx , and βy are the viscous fluxes of turbulent quantities, µt ∂k µt ∂k αx = µ + , αy = µ + , σk ∂x σk ∂y µt ∂ω µt ∂ω βx = µ + , βy = µ + , σω ∂x σω ∂y µ = µ0 /ξ1 ,
ξ1 = ρ10 /ρ0 ,
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µ0 is the molecular viscosity, Hk = Pk − Dk and Hω = Pω − Dω are the source terms, ∂u ∂v ∂v ∂u + τyy , Dk = ρkω, Pk = τxx + τxy + ∂x ∂y ∂x ∂y ω Pω = Cω1 Pk , Dω = Cω2 ρω 2 . k Here Cµ = 0.09, Cω1 = 5/9, Cω2 = 5/6, and σk = σω = 2 are the empirical constants of the model. The molecular viscosity of the mixture was calculated with Einstein’s correction. For the values of particle concentrations used in the paper, we assumed that it is possible to neglect terms that describe interaction of turbulent oscillations of particles with the carrier phase. This is partly justified by dynamic and thermal equilibrium between the phases. In the case of a heterogeneous one-velocity medium, the mean gas density is ρ1 = ξ1 ρ, where ξ1 is the dimensionless mass concentration of the gas component. After simple transformations, we obtain the following result: the equations of the k–ω turbulence model are invariant with respect to the transformation ρ1 = ξ1 ρ if the molecular viscosity obeys the law µ0 /ξ1 . Thus, the equations for turbulence parameters in the equilibrium model of mechanics of heterogeneous media are written in a manner similar to the k–ω turbulence model; the only term changed is that related to dynamic viscosity of the gas, which now has the mass concentration of the gas in the denominator. The computations were performed with the use of the present model and the model of an inviscid heatnon-conducting gas. Different coordinate systems were used: stationary and SW-fixed systems (the SW-fixed coordinate system is shown in Fig. 1). In the latter system, the wall and the gas upstream of the shock move from left to right with a velocity −D, and the velocity of the pure gas behind the SW is u2 − D. In the model taking into account viscosity and heat conduction, the following boundary conditions were set in the SW-fixed coordinate system: no-slip conditions on the wall, inviscid reflection on the upper boundary, “soft” conditions on the right boundary, and undisturbed gas conditions on the left boundary. A rectangular difference grid refined toward the wall surface was constructed for computations (details of testing this method can be found in [11, 12]). Here, we only note that the refining parameter was chosen such that the wall coordinate y + in the node closest to the wall was smaller than 2. In addition, in contrast to the two-parameter k–ε turbulence model, the k–ω model does not require introduction of near-wall functions and additional terms for computing the turbulent flow near the surface. To monitor the reliability of results for the mixture, computations were also performed on a sequence of refined grids. After the
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Fig. 2. Solution of the Riemann problem. Distributions of the medium density obtained with the use of the TVD scheme (a) and CIP1 scheme (b).
numerical data reached the steady state on two different grids, the more economical one was further used. Behind the front of the leading SW propagating in the pure gas, there arises a thin boundary layer, which does not influence the flow structure in the layer containing the mixture of the gas and solid particles affected by the incident SW. Another reason is the fact that the shock wave was placed in an immediate vicinity of the leading edge of the cloud to reduce the computation time; hence, the boundary layer did not have enough time to increase substantially. This situation models the conditions of some experiments in short shock tubes. In the inviscid problem, the reflection conditions were imposed on the vertical velocity component on the wall.
adapted in the present work to solve problems of heterogeneous flows. The CIP1 scheme includes special processing of the solution at the point where the derivatives are discontinuous. For instance, the derivatives on the left and on the right of the edge of a rectangular wave have different values, and the derivative in this case is chosen depending on the sign of velocity. At all the remaining points, except for those on the discontinuity, the left and right derivatives are assumed to be identical. The program involves a special technology for finding the discontinuity and for determining on which side of the discontinuity a particular point of the grid is located; with allowance for this fact, the derivative in this particular node is determined. The values of the functions and derivatives in the nodes at the advective stage were determined as follows: 0 fin+1 = Fi (xi − u∆t) = [(ai ξ + bi )ξ + fi,r ]ξ + fi ,
NUMERICAL METHOD. TEST COMPUTATION Let us consider some specific features of computations based on the Euler equations. Numerical implementation of this problem involved the cubic interpolation propagation (CIP) method, which was developed in [13] to improve reproduction of the behavior of contact and shock discontinuities and to solve problems with interaction of these discontinuities. The CIP scheme is based on identification of terms associated with transport in partial differential equations and approximation of this part by a cubic spline. The non-advective part is approximated by a standard difference scheme. The CIP1 method developed in [13] for solving problems of gas dynamics was implemented and
dFi (xi − u∆t) 0 = (3ai ξ + 2bi )ξ + fi,r dx for u < 0 and 0
fi n+1 =
0 fin+1 = Fi (xi − u∆t) = [(ai ξ + bi )ξ + fi,l ]ξ + fi ,
dFi (xi − u∆t) 0 = (3ai ξ + 2bi )ξ + fi,l dx for u > 0. Here ai and bi are coefficients of the spline; 0 0 fi,l and fi,r are the derivatives on the left and on the right, respectively. This procedure substantially improved the accuracy of solution matching on discontinuities. At the same time, in the class of problems associated with interaction of discontinuous fronts (shock waves and contact boundaries), the use of methods of higher-order approximation, e.g., TVD schemes, leads 0
fi n+1 =
Computation of Dust Lifting behind a Shock Wave Sliding along the Layer
Fig. 3. Estimated layer thickness versus the volume concentration of the solid phase.
to nonmonotonicity of the solution behavior on the discontinuity, and the solution becomes oscillating in the case of interacting discontinuities, whereas lowerorder schemes, which yield non-oscillating solutions, “smear” shock fronts and especially contact discontinuities. Thus, it seems important to develop a highaccuracy method that adequately describes the behavior of discontinuities. Figure 2 shows the distributions of the medium density ρ as a function of the spatial variable x, which were obtained by TVD and CIP1 methods for a test case of the Riemann problem. CIP computations are preferable for this particular problem: the number of points allocated for the fronts of the contact discontinuity and shock wave is smaller, i.e., the discontinuities are “smeared” over a smaller number of cells.
DISCUSSION OF NUMERICAL RESULTS. COMPARISON WITH EXPERIMENTS [14, 15] Let us discuss some results of numerical computations and their comparison with experiments [14, 15]. Matsui [14] considered propagation of a shock wave in air in a circular shock tube 5.2 m long. The tube walls were initially covered by a layer of soot. For this purpose, an equimolar acetylene–oxygen mixture was ignited in the high-pressure chamber, whereas the test section was filled by acetylene at an identical pressure. The diaphragm between the sections was broken at the moment of ignition of the mixture in the high-pressure
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Fig. 4. Pressure distribution over the tube surface for Ms = 1.6 and m2 = 0.001 and 0.002.
chamber. Propagation of the resultant detonation wave over the low-pressure chamber yielded solid reaction products: carbon (soot). The major part of soot deposited on the tube bottom, and the remaining soot stayed on the walls and on the upper part of the tube. The thickness of the soot layer was controlled by the initial pressure varied from 70 to 150 kPa. After that, passage of a shock wave with a Mach number Ms = 1.62 was ensured along the resultant layer of dust. The pressure at the end face of the shock tube was measured by two piezoelectric probes on the upper and lower walls, which made it possible to determine the pressure distribution on the tube walls as a function of time. The initial parameters of the problem were the layer thickness, the density of the particle material, the volume concentration of particles, and the SW Mach number. To obtain a rough estimate of the layer thickness, we assumed that soot is formed as a result of complete decomposition of acetylene located in the test section with the initial pressure equal to the atmospheric value. We estimated the soot-layer thickness as a function of concentration of the solid phase, which can be done if the mass of the solid phase is known (Fig. 3). Thus, we used the following initial conditions: Ms = 1.62, h = 4 mm, and m2 = 0.001 and 0.002. As the particle size was in the submicron range, the use of the equilibrium approach was justified. The particle density was 1900 kg/m3 . The computed pressure distributions over the tube surface for m2 = 0.001 and 0.002 are plotted in Fig. 4, which also shows the experimental
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Fig. 5. Pressure distribution at the end face of the shock tube versus time.
results. Satisfactory agreement between the experimental and numerical data is observed in the first peak of the pressure distribution caused by impingement of the incident SW onto the substrate and by the rarefaction wave resulting from SW interaction with the contact boundary between the mixture and the pure gas. In addition, the numerical solution exactly predicts the equilibrium pressure in the layer far downstream from the SW. In the case of a lower volume concentration of the solid phase, the wave period decreases; simultaneously, the wave amplitude decreases with increasing particle concentration. The computed data were also compared with the experimental results of [15]. The problem of normal incidence of the SW onto a layer of fine particles adjacent to a rigid wall was considered. The particle material was polystyrene with density ρ22 = 1060 kg/m3 and specific heat c2 = 340 J/(kg · K); the layer thickness was 20 mm, and the volume concentration of particles was m2 = 0.29. The pressure distribution on the end face of the vertical shock tube was registered as a function of time. The low-pressure chamber was filled by air with p0 = 0.1 MPa, and the pressure ratio at the diaphragm was equal to 2. The computations were performed with m2 = 0.015. The computed and experimental pressure distributions on the end face are plotted in Fig. 5. The computed and experimental data are seen to be in qualitative agreement. The first peak of pressure and the amplitude of the second oscillation are resolved by the numerical method. Certainly, we can speak only about the qualitative agreement because a layer of fine particles with a higher concentration was used in exper-
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Fig. 6. Pressure distribution on the shock-tube bottom versus time.
Fig. 7. Pressure distributions obtained by the model of [15] and by the model suggested in the present work.
iments. Possibly, this is the reason for a somewhat higher steady-state pressure in the computations than in the experiments. Gelfand et al. [15] also described the results of two-dimensional experiments on interaction of an incident SW with a dust layer lying on the shock-tube bottom. In the one-dimensional experiments, the layer was located at the end face of a vertical shock tube, and the wave propagated normal to the end face. In the two-dimensional formulation, the dust was located on the bottom wall of a horizontal shock tube, and the SW propagated along the layer; hence, the nonone-dimensionality of the interaction pattern had to be taken into account. Pressure oscillograms in the gas
Computation of Dust Lifting behind a Shock Wave Sliding along the Layer a
b
c
d
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Fig. 8. Density fields for a rectangular edge of the layer (a and c) and for a triangular edge (b and d): results of inviscid computations (a and b) and viscous computations (c and d).
Fig. 9. Pressure distribution over the surface at the time t = 0.22 msec for different shapes of the leading edge of the layer (viscous computation).
and the distribution of loading on the wall under the layer at that point of the tube were given in [15]. In the two-dimensional problem, in contrast to the onedimensional problem, the first pulse on the layer surface emerges earlier than that on the substrate: there is a delay in SW arrival, which depends on the layer thickness. Sand particles with density ρ22 = 2450 kg/m3 and
specific heat c2 = 840 J/(kg · m3 ) were used; the layer thickness was up to 25 mm. The pressure ratio on the diaphragm was p1 /p0 = 2. The initial pressure in the low-pressure chamber was 0.1 MPa, as in the previous case. The computation was performed in the twodimensional inviscid formulation. The value m2 = 0.001 was used in the computations. Figure 6 shows the computed and experimental pressure distributions over the surface. As in the one-dimensional case, the twodimensional flow displays qualitative agreement in pressure behavior in the physical and numerical experiments. It is of interest to compare our numerical data with the results obtained in [16] on the basis of the mathematical model developed in [15], which the authors claim to be valid for moderately large thicknesses of dust layers. The computations based on the model of [15] offer a satisfactory description of experimental dependences for several values of the layer thickness. The disperse phase was sand whose volume concentration in the experiment was 0.73. Numerical computations with lower values of particle concentration on the basis of the equilibrium model of mechanics of heterogeneous media provided qualitative agreement of results (Fig. 7). Certainly, the quantitative agreement of data in this case is not that representative because the values of particle concentrations in the experiment and computation were different.
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a
b
c
d
Fig. 10. Pressure fields (a and c) and density fields (b and d) at the times t = 23 (a and b) and 69 µsec (c and d).
Fig. 11. Isolines of streamwise velocity (m/sec) without allowance (a) and with allowance (b) for turbulence of the mixture.
DUST LIFTING FROM A LAYER WITH AN OBLIQUE EDGE AND REAL SW Computations were performed for SW interaction with a dust layer, the leading edge of the SW being initially inclined to the plate at an angle other than 90◦ , and the fields of parameters were compared for triangular and rectangular edges of the layer. Figure 8 shows the density fields computed on the basis of inviscid and viscous models for different shapes of the leading edge of the layer.
In the inviscid flow of the mixture (Fig. 8a and b), the main difference is the shape of the leading edge of the cloud. The height of the leading vortex in the case of the oblique edge is slightly smaller than in the case of the rectangular leading edge of the layer. This difference disappears with flow evolution, the difference in quantitative distributions of parameters is insignificant, and the flow patterns become almost identical after a certain time. In the viscous model, there is also a slight difference in the shape of the leading vortex in these two cases (Figs. 8c and 8d). There are also certain dif-
Computation of Dust Lifting behind a Shock Wave Sliding along the Layer ferences in the quantitative distributions of parameters. In the case of the oblique edge, the period of waves on the shock-tube bottom is somewhat smaller, which can be attributed to the smaller thickness of the layer (the SW enters a layer whose thickness increases up to a constant value) at the initial stage of flow evolution. The first maximum of density and pressure on the wall is higher than that in the case of the rectangular edge, as is seen from the surface-pressure plots in Fig. 9. In the general case, we also considered the influence of the rarefaction region adjacent to the SW front on lifting of the dust layer with a rectangular edge. Such shock waves are often encountered in practice and are identified with blasting. Figure 10 shows the pressure and density fields for two times. The density fields display the boundary of the dust layer and the changes in the shape of its leading edge. The SW front is more clearly seen in pressure fields. The dark region near the wall corresponds to higher pressure, which decreases in the unloading wave afterwards. The length of the rarefaction regions in the layer increases with time, and the peak pressure value decreases. The reason is that the rarefaction wave in the pure gas is levelled off, whereas the pressure in the layer increases under the action of the SW. As a result, we obtained the distributions of parameters shown in Fig. 10, which are affected by the reduced pressure in the rarefaction wave in the pure gas. By comparing the density distributions in the problems with an explosive SW and a rectangular front, we can note that the layer thickness immediately behind the shock wave is essentially unchanged, whereas the layer is significantly compressed in the case of the rectangular front. In this case, the SW intensity rapidly decreases under the action of an attached unloading wave, and the layer experiences a weaker action.
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Fig. 12. Pressure distribution over the plate surface with and without allowance for turbulence for Ms = 2.5 and m2 = 0.0012.
a
b
EFFECT OF TURBULENCE To elucidate the influence of turbulence of the mixture on the flow pattern behind the leading SW, we performed computations based on the model of a viscous heat-conducting mixture with an original technique based on the TVD method [12]. The computations were performed with and without allowance for turbulence of the mixture. Figure 11 shows the isolines of streamwise velocity of the mixture for these two cases with the initial volume concentration of particles in the layer m2 = 0.001 and the SW Mach number Ms = 1.6. As in the previous computations of flows behind shock waves entering the layer, an oblique SW is formed in the layer; this SW is reflected from the substrate and reaches the contact boundary. The contact boundary is
Fig. 13. Density fields computed without allowance (a) and with allowance (b) for turbulence for Ms = 1.6 and m2 = 0.0057.
substantially shifted toward the wall and uprises in the rarefaction wave. This favors spraying of the layer. It is seen from Fig. 11 that the basic qualitative difference in the behavior of the turbulent and non-turbulent flows of the mixture is the shape of the leading edge of the layer. If turbulence is taken into account (see Fig. 11b), a thin high-velocity trickle is formed near the wall; its presence
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a
b
cause of propagation of stronger shock waves.The internal shock waves in the turbulized layer are weaker because of dissipation and participate in particle lifting at a greater distance downstream from the leading SW. This is expressed in more significant lifting of the dust layer. Figure 14 shows the density fields computed with allowance for turbulence and the shadowgraph obtained in the experiment of [17]. The results are in qualitative agreement. Indeed, the flow pattern visualized in computations has a vortex on the leading edge, which is also observed in the experiments. CONCLUSIONS
Fig. 14. Density field calculated (Ms = 1.6 and m2 = 0.001) with allowance for turbulence (a) and shadowgraph obtained in the experiments of [17] (b).
can be explained by higher shear stresses in this region owing to turbulent viscosity. The leading edge of the layer in a non-turbulent flow remains almost vertical. These types of the flow can be quantitatively compared on the basis of the pressure distributions over the plate surface in Fig. 12. The data were obtained for Ms = 2.5 and m2 = 0.0012. In computations with allowance for turbulence of the mixture, the period of internal waves is larger and their amplitude is lower. The reason is additional turbulent viscosity, which makes the SW deflect from the normal position by a greater angle than in the case of the non-turbulent flow. Thus, the distance between the first and subsequent points of incidence of internal shock waves on the wall increases, and the SW intensity decreases because of additional dissipation. Figure 13 shows the density fields for a denser layer. The intensity is higher in the absence of turbulence because of lower dissipation. In the turbulent flow, the number of SW reflections from the substrate (periods of interaction of internal waves) is greater and lifting of particles by the leading vortex is more significant, which is reflected in density distributions in the medium. Within the framework of the non-turbulent approach, the layer of particles remains pressed to the surface be-
A mathematical model of mechanics of an equilibrium heterogeneous medium with allowance for turbulence of the mixture has been developed to describe dust lifting from a layer under the action of a rectangular or triangular leading SW. To solve this problem of mechanics of heterogeneous media in an inviscid formulation, the CIP method has been modified and tested, which allows one to correctly solve problems of mechanics of heterogeneous media with curvilinear contact boundaries. The mathematical model proposed for the description of the dust-layer flow under the action of an SW has been verified by comparisons with experimental dependences of pressure on the substrate versus time in two- and one-dimensional geometry. It has been demonstrated that a high-velocity nearwall trickle is formed in the turbulent mixture at the leading edge of the layer, and dust is lifted more intensely owing to internal shock waves attenuated by dissipation. The authors are grateful to N. N. Fedorova for assistance in performing this work. This work was supported by the Russian Foundation for Basic Research (Grant No. 03-01-00453) and by the Ministry of Education of the Russian Federation (Foundation for Supporting Research Activities of Post-Graduate Students, Grant No. A03-2.10-544).
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