vladimir v. semionov a and alexey n. volkov...

S3_Fri_B_67 6th International Conference on Multiphase Flow, ICMF 2007, Leipzig, Germany, July 9 – 13, 2007

1

Effect of solid particles on flow structure of supersonic two-phase gas-solid particles

impact jet in the self-oscillation regime

Vladimir V. Semionov a and Alexey N. Volkov b, *

a Baltic State Technical University, Aerospace Faculty, Department of Plasma- and Gas Dynamics 1, 1-ya Krasnoarmeiskaya ul., Saint-Petersburg, 190005 Russia

E-mail: [email protected]

b University of Virginia, Material Science and Engineering Department 117 Engineer's way, Charlottesville, VA 22903, USA

E-mail: [email protected]

* Corresponding author

Keywords: supersonic two-phase impinging jet, two-way coupling effects, particle-particle collisions, self-sustained oscillations

Abstract A flow in a supersonic gas-solid impinging jet is studied numerically. The unsteady axisymmetric flow of the carrying gas is described by the Euler equations with additional terms which account for the two-way coupling effects. The kinetic model developed by Volkov et al. (2005) is used in order to describe motion and heat transfer of solid particles and inelastic particle-particle collisions. The flow of the particulate phase is calculated by the direct simulation Monte Carlo method. The numerical method for carrying gas flow is based on the Harten’s TVD scheme. Multi-block computational grid adapted to the structure of the jet is used in simulations. Computed flow structure in the single-phase jet is compared with experimental data obtained by Semiletenko & Uskov (1972) and a good agreement is established for both the stationary and self-oscillations regimes. In calculations of two-phase jets, the size of solid particles and the concentration of the particulate phase are varied. The patterns of modelling particles of various radii in the regime of self-sustained oscillations are obtained and analyzed. It is found that three qualitatively different regimes of the particle phase flow can take place. The influence of particles on the carrying gas flow results in damping of flow oscillations. For a given mass concentration of particles, the amplitude of oscillation drops with decrease in particle radius.

Introduction The analysis of flows in multi-phase supersonic jets interacting with obstacles are of great importance for understanding of processes in solid propellant boosters and for industrial applications where multi-phase jets are used for cutting of materials and for coating of surfaces. Several qualitatively different flow regimes can be realized even in the simplest case when the jet interacts with a perpendicular flat obstacle (Ginzburg et al., 1976). In particular, in some range of governing parameters the self-oscillation regime exists when the whole flow between the nozzle and the obstacle experiences self-sustained quasi-periodic oscillations, e.g. see Yasunobu et al. (2005). The effect of solid particles on the carrying gas flow in supersonic gas solid-particle jet was studied by Ishii et al. (1989). The motion of solid particles in a subsonic impinging jet was considered by Kitron et al. (1988), who emphasized the effect of collisions between solid particles on the energy transfer from the jet to the obstacle and on the erosion of the obstacle. For the authors’ knowledge, flow of supersonic multi-phase impinging jets in the regime of self-sustained oscillations has not been studied numerically yet. In this work, the flow in a supersonic gas-solid impinging

(or impact) jet with a perpendicular obstacle is considered. The purpose of the paper is to study regimes of the particle phase flow and the effect of solid particles on the jet structure in the regime of self-sustained oscillations. This work is focused on flow conditions when turbulence and viscosity of the carrying gas can be neglected. Neglecting of turbulence and viscosity restricts the range of governing parameters under consideration. At the same time flow in the supersonic single-phase impinging jet with self-sustained oscillations is studied successfully on the base of the Euler equations by Kuzmina & Matveev (1979) and by Adrianov et al. (2000). We believe that the model based on Euler equations can also predict the effect of solid particles on oscillations in two-phase impinging jets. Nomenclature t time (s) x, y cylindrical coordinates (m) da diameter of the nozzle exit (m) L distance from the nozzle exit

to the obstacle (m) V vector of translational velocity (m·s-1)

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p pressure (Pa) T temperature (K) rp particle radius (m) M Mach number St Stokes number сp mass concentration of particles np numerical concentration of particles (m-3) F frequency of oscillation (Hz) Greek letters ρ density (kg·m-3) μ viscosity (kg·m-1·s-1) γ ratio of heat capacities (adiabatic index) τ period of oscillation (s) Subsripts a parameters at the nozzle exit e parameters of the ambient gas i parameters of an individual solid particle p parameters of the particle phase Formulation of the problem and basic assumptions We consider a supersonic gas-solid particle impinging jet flowing from the circular nozzle OF (Fig. 1) with diameter da into an ambient gas with the pressure pe and the temperature Te. The gas, flowing from the nozzle, and the ambient gas are assumed to have the same chemical composition and physical properties. The finite thickness of the nozzle edge EF is taken into account. The jet impacts the flat obstacle AB which is placed at the distance L from the nozzle exit. The obstacle is perpendicular to the axis of the jet.

The flow at the nozzle exit OF is assumed to be uniform with given pressure pa, temperature Ta and velocity Va. Particles are introduced into the flow only through the nozzle exit where they are in mechanical and thermal equilibrium with surrounding gas. In the single-phase supersonic under-expanded impinging jet, at least five different flow regimes are observed in experiments, see Ginzburg et al. (1976). In this study, only

two of them are considered. First one corresponds to the steady-state flow without a vortex ring near the obstacle. A typical structure of the jet in this case is depicted in Fig. 1. The second flow regime is the regime of strong self-sustained oscillations in the jet flow. In this case the flow has the same structure as in the previous case, but positions of shock waves are changed in time quasi-periodically and, in consequence, the whole jet structure is affected by strong oscillations. The mathematical model of the flow is based on the following assumptions: • The flows of both phases are axisymmetric. • The turbulence generation is negligible and the

carrying gas can be treated as an inviscid fluid. • Solid particles are spheres of radius rp. • The “gas” of particles is rarefied and its flow can be

described by the kinetic model based on the Boltzmann-type kinetic equation, where only binary collisions are taken into account. The hydrodynamic interaction between particles is negligible.

• The reverse action of particles on the carrying gas can be determined as a sum of actions of individual particles.

The axisymmetric flow is considered in the cylindrical coordinates x, y, where axis Ox is the axis of flow symmetry and y is the distance from the axis of symmetry (see Fig. 1). For numerical study of the considered problem the model proposed by Volkov et al. (2005) is used. This combined model consists of the kinetic model for solid particles and the macroscopic model for carrying gas flow. The two-way coupling effects are taken into account by means of additional terms in the governing equations of the carrying gas flow. These terms are calculated by averaging of parameters of individual solid particles. This model was applied to the study of stationary supersonic gas-solid flow over blunt bodies in (Volkov & Tsirkunov, 2002; Volkov et al., 2005). In this paper, the model is adopted for numerical simulation of unsteady axisymmetric gas-solid flows. Further this model is described only briefly. Full description of the model can be found elsewhere. The kinetic model for solid particles It is assumed that the state of any ith particle at the time t is described by its radius-vector ri, vector of translational velocity Vi, vector of rotational velocity ωi and the uniform temperature Ti of the particle material. An inelastic binary collision between a pair of particles is considered as a collision of rotating hard spheres with total cross-section 4πrp2. Parameters of ith and jth particles after their collision can be found as follows (Oesterle & Petitjean, 1993; Volkov et al., 2005)

p

ii mJVV += −+ ,

pjj m

JVV −= −+ , (1)

Jnωω ×+= −+p

pkk I

r,

ppkk cm

KTT2

Δ+= −+ , 2,1=k ,

( )⎥⎦

⎤⎢⎣

⎡⋅−

−+⋅

+= −−− nnuunnuJ )(

21

)(2

1 ptpnp

aam ,

)( −−−−− +×+−= jipij r ωωnVVu ,

Obstacle

Mach waveNozzle exit

Jet boundary

O A

BC

D E

Fda/2

L x

y

Triple point

Figure 1: Sketch of the computational domain ABCDEFO and typical flow structure of a supersonic impinging jet.

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where the superscripts “–” and “+” denote the particle parameters before and after the collision, mp and Ip are the mass and the moment of inertia of a spherical particle, cp is the specific heat of the particle material, ΔK is the difference in the kinetic energy of the pair of particles before and after the collision, n is the unit vector directed from the center of ith particle to the center of jth particle, coefficients apn and apt are the parameters of the model which account for inelasticity of collisions and friction between particles’ surfaces. The flow of the particle phase is described by the one-particle distribution function f = f ( ri, Vi, ωi, Ti, t ) which is normalized by the numerical concentration of particles np

∫= iiiiiip dTddtTftn ωVωVrr ),,,,(),( .

The particle distribution function f can be found as a solution of the Boltzmann-type kinetic equation

+⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

+⋅∂∂

+∂∂ f

If

mf

tf

p

i

ip

ii

i

lω

fVr

)(

)( fIfcm

qc

pp

i =⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

+ ,

where fi and li are the aerodynamic force and torque exerted on ith particle by the carrying gas, qi is the heat flux on the particle surface and Ic(f) is the collisional integral. The structure of Ic(f) is described in (Volkov et al., 2005). The macroscopic quantity [Φ] of the particle phase, which is defined as the total value of physical quantity Φ per unit volume of the gas-particle mixture, can be expressed as a statistical averaged value of the corresponding quantity Φ = Φ ( ri, Vi, ωi, Ti ) of an individual particle

∫Φ=Φ iiiiiiiii dTddtTfTt ωVωVrωVrr ),,,,(),,,(),]([ . (2) For example, if E = mpVi2/2 + Ipωi2/2 + U(Ti) is the full energy of ith particle (U(Ti) is the internal energy of the particle), then [E] is the full energy of the particle phase per unit volume. Particles are introduced into the computational domain through the nozzle exit OF where the particle numerical concentration npa is specified and the inflow boundary conditions is defined as follows at 0>ixV : )()ω()()()( aiiiziyaixpa TTVVVVnf −−= δδδδδ . (3) Here Vix, Viy and Viz are the components of the vector Vi. The inelastic collision of a particle with a solid boundary of the computational domain is described by the semi-empirical model proposed by Tsirkunov et al. (1994) for plane collision and generalized by Volkov et al. (2005) for the case of arbitrary directions of translational and rotational velocities of the particle during the collision with the wall. The ambient gas is assumed to be a single-phase fluid so that the inflow of solid particles through boundaries BC and CD is absent.

Governing equation and boundary conditions for carrying gas flow The flow of the carrying gas is described by the Euler equations with additional terms which account for the gas-particle interaction. For a 2D axisymmetric flow these equations can be written in the conservative form as follows (Toro, 1999)

HDGFU =+∂∂

+∂∂

+∂∂

yyxt1 , (4)

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

y

x

VV

e

ρρ

ρρ

U ,

( )

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

+

=

yx

x

x

x

VVpV

VpeV

ρρ

ρρ

2F ,

( )

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

+

=

pVVV

VpeV

y

yx

y

y

2ρρρρ

G ,

( )

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ +

=

2y

yx

y

y

VVV

VpeV

ρρρρ

D ,

RTp ρ= , 21

22yx VVTRe

++

−=

γ,

where ρ, p, T and e are the density, the pressure, the temperature and the full specific energy of the gas; Vx and Vy are the components of the gas velocity vector V in the cylindrical coordinates x, y; R is the specific gas constant and γ is the ratio of gas heat capacities. Source term H in Eq. (4) describes the effect of particles on the carrying gas flow. It was assumed that the Mach number at the nozzle exit is not less than unity. In these conditions the inflow boundary conditions at the nozzle exit OF are applied:

aTT = , app = , ax VV = , 0=yV .

At the solid surfaces the no-penetration boundary conditions are used:

at EF and AB: 0=xV , at ED: 0=yV .

The boundary conditions at BC and CD were different depending on the instant value of a parameters Mα which are similar to the local Mach number

RTVtMγ

αα =),(r , yx,=α

at these boundaries. In regions of subsonic flow, where the gas velocity points inside the domain (inflow boundaries), the gas pressure and temperature are assumed to be equal to the pressure pe and the temperature Te of the ambient gas:

at BC if 01


4

In regions of subsonic flow where the gas velocity points outside the domain (outflow boundaries) only the gas pressure is assumed to be equal to pe, i.e.

at BC if 10


5

Numerical Scheme For calculations, the combined CFD/DSMC numerical method developed by Volkov & Tsirkunov (2002) is used. In this study this method was adopted for calculations of 2D axisymmetric unsteady supersonic gas-particle flows. Particle phase is represented as a set of large number of modeling particles. At every time step Δt of the unsteady process the computations include three stages:

1. Calculation of motion of modeling particles, particle-particle and particle-wall collisions and generation of new particles at the nozzle exit.

2. Calculation of the term H in Eq. (4), describing the action of solid particles on the carrying gas flow, and the macroscopic parameters of particle phase at this time step.

3. Numerical solution of equations (4) for carrying gas flow.

On stage 1 the direct simulation Monte Carlo method (DSMC) for solid particles is used. This method was originally developed by Volkov & Tsirkunov (1996) on the basis of the majorant frequency scheme proposed by Ivanov & Rogazinskii (1988) for rarefied gas flows. On stage 2 calculations of the gas-particle interaction term H and the macroscopic parameters of the particle phase flow are performed by the spatial averaging of parameters of modeling particles over the volume of a cell of the computational grid instead of ensemble averaging implying in Eq. (2). On stage 3 the numerical algorithm for solution of the modified Euler equations (4) is based on the Harten’s TVD scheme (Yee & Harten, 1988). Typical number of modeling particles in a cell of the computational grid inside the jet region was equal to ≈20. It is not enough to eliminate a statistical noise in calculations of the gas-particle interaction terms H and the particle phase macroscopic parameters. Therefore, the computational results for two-way coupling flows contain some statistical scatter which has numerical origin. In was found that the computation of flow parameters in the region at x < 0 (see Fig. 1) is essential for correct prediction of the flow structure of the impinging jet in the regime of self-sustained oscillations. Boundary conditions at boundaries BC and CD of this region are partially artificial and in order obtain a good agreement with experimental data these boundaries should be places far enough from the nozzle exit OF. At the same time sizes of cells of the computational grid inside the jet flow between the nozzle exit and the obstacle must be small enough in order to obtain a good accuracy in representation of the complex unsteady structure of the shock waves and the contact surfaces in this region. In order to satisfy these contradictory requirements, the multi-block computational grid is used (Fig. 2). Usually it contains four sub-grids I, II, III, and IV. Computational sub-grid I is uniform and fine, spacing between nodes in other sub-grids increases rapidly with approaching external boundaries BC and CD. The same computational grid is used for sampling of particle-particle collisions, calculation of the particle phase macroscopic parameters and for numerical solution of the governing equations for the carrying gas flow.

(a)

-20 -10 00

10

20

30

40

50

60

70

A

BC

D

Obstacle

Nozzle exit

(b)

-2 -1 0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

I

II

III

IV

Nozzle exit

Obstacle

A

E

F

Figure 2: Sketch of the multi-block computational grid. (a), structure of the whole grid; (b), structure of the computational grid in the near-nozzle region. In calculations, the sizes of the computational domain in x- and y-directions were chosen to be equal to 15da and 40da, respectively. Typical number of cells in the computational domain was equal to 5x104 and typical number of modeling particles was equal to 106. Comparison of computational results and experimental data for the supersonic single-phase impinging jet In order to verify the developed numerical algorithm and the computational code, the flow in supersonic single-phase two-phase jet was studied and computational results were

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compared with experimental data obtained by Semiletenko & Uskov (1972) for the stationary and oscillatory regimes. In Fig. 3 the experimental shadowgraph (a) is compared with the computed field of gas density (b) which was obtained for the same values of governing parameters in the case of the stationary regime when the distance L between the nozzle exit and the obstacle is equal to 2.5da.

(a)

(b)

dahm

ht

Figure 3: Experimental shadowgraph (a, Semiletenko & Uskov, 1972) and computed field of gas density (b) for the single-phase impinging jet in the stationary regime. Ma=1, рa / pe=9.3, L / da=2.5. One can see that the computed structure of the jet is in a good qualitative agreement with the structure observed in the experiments. In order to compare experimental and computational results quantitatively, the distances between the obstacle and the Mach wave at the axis of symmetry hm and between the obstacle and the triple point ht (see Fig. 3, (b)) were calculated. Semiletenko & Uskov (1972) proposed the semi-empirical formulae for these distances in the form

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−=

nMdL

nMdL

nMdh

aaaaaa

m

γγγ73.1exp83.0745.0 ,

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−=

nMdL

nMdL

nMdh

aaaaaa

t

γγγ75.1exp88.072.0 ,

where n=pa/pe.

For L / da=2.5 one can calculate from these formulae hm / da = 0.72 and ht / da = 0.98. At the same time values hm / da = 0.675 and ht/da = 0.925 were found as a results of simulation. The difference between experimental and computational values is less than 6%. In Fig. 4 the experimental (curve 1) and computed (curve 2) distributions of the gas pressure along the obstacle surface are plotted for the stationary flow regime. Agreement between them is satisfactory. With increase in the distance L between the nozzle exit and the obstacle the flow in the impinging jet becomes unstable (Ginzburg et al., 1976) and the regime of self-sustained oscillations is realized when the whole structure of the jet changes quasi-periodically. In particular, the Mach wave moves between the nozzle exit and the obstacle and the pressure distribution at the obstacle surface varies periodically. Semiletenko & Uskov (1972) found that for L/da=3.25 the minimal and maximal distances hm between the Mach wave and the obstacle at the axis of symmetry are equal to 1.85 and 2.25, respectively. Computational values of these parameters are found to be equal to 1.95 and 2.25. Therefore, the calculated amplitude of oscillation of the Mach wave position is in a good agreement with the experiment. Experimental and computed plots of the gas pressure in the stagnation point at the obstacle surface are shown in Fig. 5. The main frequency F of pressure oscillations in the stagnation point (measured as F=1/τ, where τ is the time between two sequential maximums of pressure) was found to be equal to 3000 Hz for the experimental curve and 3087 Hz in calculations. The shapes of curves also seem to be similar, but the experimental graph is rather indistinct and it is not possible to come to an absolute conclusion on this matter. Computational results show that parameters of the numerical method (cell sizes and positions of the external boundaries) have a week influence on the frequency of pressure oscillations F in the impinging jet in the regime of self-sustained oscillations. At the same time the amplitude of pressure oscillations in the stagnation point and the positions of the shock waves depend on the numerical

Distance form the stagnation point

Pres

sure

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

Experiment (Semiletenko & Uskov, 1972)Computation

p*

y/(da/2)

2

1

Figure 4: Distributions of the dimensionless pressure p*=p/pe along the obstacle surface in the stationary regime. Curve 1, experiment by Semiletenko & Uskov (1972); 2, computational results. Ma=1, рa / pe=9.3, L / da=2.75.

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parameter and for agreement between experiments and calculations a correct choice of the numerical parameters is necessary.

(a)

(b)

Time

Pre

ssur

eat

the

stag

natio

npo

int

0 20 40 60 80 1002

2.2

2.4

2.6

2.8

p*

t* Figure 5: Experimental (a, Semiletenko & Uskov, 1972) and computed (b) graphs of the time-dependant pressure p*=p/pe in the stagnation point at the obstacle in the regime of self-sustained oscillations. t*=tVa/(da/2). Ma=1, рa/pe=9.3, L/da=3.25. Experimental frequency of pressure oscillation F=3000 Hz, computed frequency F=3087 Hz. Flow patterns of solid particles in the one-way coupling flow

Contours of constant local Mach number for different times during one period of oscillations in the single-phase jet are plotted in Fig. 6 for L/da=3.25. In Fig. 7 corresponding patterns of inertialess particles (markers) are shown. The instant velocity of such a marker at every time is equal to the gas velocity in the same point. Therefore, markers represent the real motion of fluid “particles” in a non-stationary flow. In particular, patterns of markers indicate formation of a vortex near the obstacle at some distance from the stagnation point (Fig. 7, (b)). Afterwards this vortex moves along the surface to the exit boundary (Fig. 7, (c) and (d)). The oscillatory gas flow depicted in Figs. 6 and 7 was used in order to study the patterns of solid particles of various radii in the case when their concentration is small enough so that the effect of particles on the gas flow and collisions between particles are negligible. The patterns of solid particles with radius rp=1x10-6 m are

plotted in Fig. 8 for the same times during one period of oscillation as in corresponding fields in Figs. 6 and 7. The flow structure of the particle phase is unsteady and extremely complex in this case. Two main peculiarities of this flow can be observed. First one is the unsteady region with increased particles concentration behind the Mach wave (Fig., 8, (b)). This region moves towards the obstacle (Fig. 8, (c)), “falls” at the surface (Fig., 8, (d)) and completely disappears for some time during the period of oscillation (Fig. 8, (e) and (f)). Formation and motion of this region is originated by the displacement of the Mach wave. This region exists during time when the Mach wave moves towards the obstacle (see Fig. 7, (b) – (d)). Another peculiarity in the particle phase pattern is observed in the region where the layer closed to the jet boundary attaches to the obstacle (see Fig. 6). In this region a vortex in the carrying gas flow is placed (see Fig. 7) and particles, involving into the vortex flow, form a thin layer with very high particle concentration. The end of this layer is attached to the surface and it is clearly seen in Fig. 8, (a), (e) and (f). One can conclude from Fig. 8, that the distribution of particles with radius rp=1x10-6 m near the obstacle changes qualitatively several times during the period of oscillation. The patterns of the particle phase flow for different particle radii are compared in Fig. 9. Three different regimes of particle phase flow can be realized in the oscillatory supersonic impinging jet depending on particle radius rp. Fine particles with small Stokes number (e.g., rp=0.1x10-6 m, St=0.9x10-2, Fig. 9, (a)) follow the trajectories of fluid “particles” of the carrying gas flow. The flow pattern of such particles is similar to the density field of the carrying gas. In particular, the particle concentration rises abruptly just behind the Mach wave. Fine particles do not fall at the surface of the obstacle and the role of particle-particle collisions is negligible in this case. Large particles (e.g., rp≥10x10-6 m, St≥0.9x102, Fig. 9, (c) and (d)) do not follow the carrying gas flow. They move almost directly to the obstacle and can experience multiple collisions with its surface. Flow field of such particles is almost stationary in a large region around the stagnation point. The maximal energy flux density from the particle phase to the obstacle (due to the loss in the kinetic energy of particles after their inelastic reflections from the surface) is realized in the stagnation point in this case. Collisions between particles moving towards and from the obstacle change noticeably the flow fields of both phases in this case. The flow field of particles with intermediate radius (e.g., rp=10-6 m, St=0.9, Fig. 9, (b)) is most complex and, as it was shown above, is a subject of qualitative changes during a period of oscillation. Such particles can reach the obstacle surface, but particles moving near the axis of symmetry are decelerated significantly after the Mach wave. They reach the obstacle with relatively small velocities. At the same time particles moving close to the jet boundary can reach the obstacle with large velocities. Therefore, the maximal energy flux from the particle phase to the obstacle surface is realized at some distance from the stagnation point. The effect of solid particles on the flow structure in the regime of self-sustained oscillations

Computational results which are obtained for finite concentration of particles with taking into account two-way

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coupling effects and particle-particle collisions are shown in Figs. 10 – 13. If concentration of particles of intermediate radius (rp=10-6 m, Figs. 10-11 and 12, (a)) as low as cpa=2.85x10-3 the flow structure of both phases is almost the same as in one-way coupling flow at cpa→0. In this case the maximal pressure on the surface is realized at some distance from the stagnation point. Such pressure distribution is a typical feature of the flow with self-sustained oscillations in the impinging jet (Semiletenko & Uskov, 1972). The increase in the particle concentration results in decay of oscillations in the flow structure of both phases and this “lateral” maximum of pressure disappears. At the same time particles do not change the frequency of oscillation. For cpa=0.22 the maximal pressure at the surface is realized in the stagnation point (curve 4 in Fig. 10), pressure distribution along the obstacle surface is qualitatively similar to the pressure distribution in the stationary single-phase jet (see Fig. 4) and the amplitude of pressure oscillation in the stagnation point drops in approximately ten time as compared with the single-phase jet (Fig. 12, (a)). For this concentration the flow fields of both phases are almost stationary and they are shown in Fig. 11. One can see that the flow pattern of solid particles in this case is similar to the flow pattern in Fig. 8, (a). The peculiarity in the particle concentration field near the region, where the jet attaches the obstacle surface, is even more pronounced as compared with the flow in the jet with negligibly small particle concentration. Particles of considered size collide with the obstacle surface with relatively low velocities and the efficiency of collisions between particles moving towards and from the surface is low in this case. The flow field of large particles changes qualitatively due to inter-particle collisions when particle concentration is high enough (compare Figs. 12 and 9, (c)). In the collisionless flow the maximum of particle concentration is realized at some distance from the obstacle surface where the envelope of trajectories of particles after their reflection from the surface is located (Tsirkunov et al., 2002). In collision-dominated flow there is no distinct boundary of the region, where particle reflected from the obstacle are present. The maximum of particle concentration is realized at the obstacle surface in this case. Such effect of particle-particle collisions on the particle phase flow structure is similar to the effect of inter-particle collisions in two-phase gas-solid flows over blunt bodies, which was studied by Volkov & Tsirkunov (1996; 2002) and by Volkov et al. (2005). For large particles both the time-average pressure and the amplitude of pressure oscillation in the stagnation point are larger as compared with the case of particles of intermediate size (compare corresponding curves in Fig. 12, (a) and (b)). Thus, the decay of self-sustained oscillations in the supersonic two-phase impinging jet with particles of the intermediate size is more “efficient” than in the case of particles of larger size. Computational results show, if particle radius rp is less than 3x10-6 m, then the almost full damping of self-sustained oscillations in supersonic two-phase impinging jet takes place at cpa≥0.2. For particles with larger radius (rp≥ 3x10-6 m) the full damping of oscillations is not observed in the considered range of particle concentration.

Conclusions The numerical algorithm was developed for modelling of supersonic two-phase impinging jets for both stationary and oscillatory flow regimes. Flows in the single-phase impinging jets were calculated in order to verify the numerical model. Computational results are found to be in a quantitative agreement with experiments by Semiletenko and Uskov (1972). Supersonic flows in two-phase impinging jets are studied numerically in wide ranges of particle size and solid phase concentration at the nozzle exit. In was found that three qualitatively different regimes of particle flow take place depending on the particle size. Interaction of particles with the unsteady vortex flow of the carrying gas results in formation of some peculiarities in the particle concentration field near the obstacle. Increase in the particle concentration results in gradual decay of oscillations in the flow structures of both phases while the frequency of oscillation remains unchanged. For a given mass concentration of particles, the effectiveness of damping of oscillations decreases with increase in the particle radius. Acknowledgements This work was supported by the Grant Center for Natural Sciences at the Saint-Petersburg State University (project No. A03-2.10-221) and by the International Science and Technology Center (project No. 3026). Authors would like to thank Prof. Yu.M.Tsirkunov for helpful discussions of some aspects of the study. References

Adrianov, A.L., Bezrukov, A.A. & Gaponenko, Ya.A. Numerical modeling of interaction between a supersonic jet and a flat obstacle. Prikladnaya mekhanika I tekhnicheskaya fizika. Vol. 41, No. 4, 106 – 111 (2000) [in Russian] Dennis, S.C.R., Singh, S.N. & Ingham, D.B. The steady flow due to a rotating sphere at low and moderate Reynolds numbers. J. Fluid Mech., Vol. 101, 257 – 279 (1980) Ginzburg, I.P., Sokolov, E.I. & Uskov V.N. Types of the flow structure in the interaction between under-expanded jet and infinite flat obstacle. Prikladnaya mekhanika I tekhnicheskaya fizika, No. 1, 45 – 50 (1976) [in Russian] Henderson, C.B. Drag coefficients of spheres in continuum an rarefied flows. AIAA J., Vol. 14, 707 – 708 (1976) Ishii, R., Umeda, Y. & Yuhi, M. Numerical analysis of gas particle two-phase flows. J. Fluid. Mech., Vol. 203, 475 – 515 (1989) Ivanov, M.S. & Rogazinskii, S.V. Comparative analysis of DSMC techniques in rarefied gas dynamics. Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki, Vol. 28, 1058 – 1070 (1988) [in Russian]

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Kitron, A., Elperin, T. & Tamir, A. Monte Carlo analysis of wall erosion and direct contact heat transfer by impinging two-phase lets. J. Thermophys., Vol. 3, 112 – 122 (1988) Kuzmina, V.E. & Matveev, S.K. About a numerical study of unstable interaction between a supersonic jet and a flat obstacle. Prikladnaya mekhanika I tekhnicheskaya fizika. No. 6, 93 – 99 (1979) [in Russian] Oesterle, B. & Bui Dinh, T. Experiments on the lift of a spinning sphere in a range of intermediate Reynolds numbers. Exp. Fluids, Vol. 25, 16 – 22 (1998) Oesterle, B. & Petitjean, A. Simulation of particle-to-particle interactions in gas–solid flows. Int. J. Multiphase Flow, Vol. 19, 199 – 211 (1993) Rubinow, S.I. & Keller, J.B. The transverse force on a spinning sphere moving in viscous fluid. J. Fluid Mech., Vol. 11, 447 – 459 (1961) Semiletenko, B.G. & Uskov, V.N. Experimental correlations for the positions of shock waves in a jet flowing to a perpendicular obstacle. Inzhenerno-Fizicheskii Zhurnal, Vol. 23, 453 – 458 (1972) [In Russian] Sternin, L. E., Maslov, B.N., Shraiber, A.A. & Podvysotskii, A.M. Two-phase mono- and polydisperse gas-particle flows, Mashinostroenie, Moscow (1980) [in Russian] Toro, E. F. Riemann solvers and numerical methods for fluid dynamics. Springer, Berlin (1999) Tsirkunov, Yu.M., Panfilov, S.V. & Klychnikov, M.B. Semiempirical model of impact interaction of a disperse impurity particle with a surface in a gas suspension flow. J. Engng. Phys. Thermophys., Vol. 67, 1018 – 1025 (1994) Tsirkunov, Yu.M., Volkov, A.N. & Tarasova, N.V. Full Lagrangian approach to the calculation of dilute dispersed phase flows: advantages and applications. In: CD-ROM Proc. ASME FEDSM_02, Montreal, Canada, paper 31224 (2002) Volkov, A.N. & Tsirkunov, Yu.M. Direct simulation Monte-Carlo modelling of two-phase gas–solid particle flows with inelastic particle–particle collisions. In: Proc. Third ECCOMAS Comp. Fluid Dynamics Conf., Paris, France, 662 – 668 (1996) Volkov, A.N. & Tsirkunov, Yu.M. CFD/Monte Carlo simulation of collision-dominated gas–particle flows over bodies. In: CD-ROM Proc. ASME FEDSM_02, Montreal, Canada, paper 31222 (2002) Volkov, A.N., Tsirkunov, Yu.M., & Oesterle, B. Numerical simulation of a supersonic gas–solid flow over a blunt body: The role of inter-particle collisions and two-way coupling effects. Int. J. Multiphase Flow, Vol. 31, 1244 – 1275 (2005) Yasunobu, T., Otobe, Y., Kashimura, H. & Setoguchi, T. Characteristics of oscillation frequency caused by supersonic impinging jet. CD ROM Proc. Int. Conf. on Jets, Wakes and

Separated Flows, October 5-8, 2005, Toba-shi, Mie, Japan (2005) Yee, H.C. & Harten, A., 1987. Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates. AIAA J., Vol. 25, 266 – 274 (1987)

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Figure 6: Contours of constant local Mach number M=|V|/(γRT)1/2 for the single-phase impinging jet in the regime of self-sustained oscillations at different time t during a period of oscillation. (a), t/τ = 0; (b), t/τ = 0.2; (c), t/τ = 0.4; (d), t/τ = 0.6; (e), t/τ = 0.8; (f), t/τ = 1. Ma=1, рa / pe=9.3, L / da=3.25, period of oscillation τ = 3.2x10-4 s.

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Figure 7: Patterns of inertialess particles (markers) in the single-phase impinging jet in the regime of self-sustained oscillations at different time t during a period of oscillation. (a), t/τ = 0; (b), t/τ = 0.2; (c), t/τ = 0.4; (d), t/τ = 0.6; (e), t/τ = 0.8; (f), t/τ = 1. Ma=1, рa / pe=9.3, L / da=3.25, period of oscillation τ = 3.2x10-4 s.

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Figure 8: Patterns of solid particles in the impinging jet in the regime of self-sustained oscillations at different time t during a period of oscillation. (a), t/τ = 0; (b), t/τ = 0.2; (c), t/τ = 0.4; (d), t/τ = 0.6; (e), t/τ = 0.8; (f), t/τ = 1. Ma=1, рa / pe=9.3, L/da=3.25, period of oscillation τ = 3.2x10-4 s. Particle concentration is assumed to be negligible so that the effect of particles on the gas flow and collisions between particles are not taken into account.

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Figure 13: Flow pattern of solid particles with radius rp=10x10-6 m. Ma=1, рa / pe=9.3, L / da=3.25, cpa=0.22.

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