analysis of unsteady supersonic two-phase flows by the particle-in-cell method

Computers and Fluids, Vol. 3, pp. I 11-123. Pergamon Press, 1975. Printed in Great Britain

ANALYSIS OF UNSTEADY SUPERSONIC TWO-PHASE FLOWS BY THE PARTICLE-IN-CELL METHOD

ANDREW S. LEVINE

Raytheon Company, Bedford, Massachusetts, 01730, U.S.A.

and

BERNARD OTTERMAN

Hofstra University, Hempstead, N.Y. 11550, U.S.A.

(Received 26 March 1973 ; revised 15 June 1974)

Ahstract--A procedure for analyzing the unsteady flow of gas-particle suspensions, based on the Particle-In-Cell (P1C) method, is presented. Interactions between the phases are incorporated in the Lagrangian portion of the P1C calculation, while discrete 'mass points' are used to represent each of the phases of the suspension during the material transport. Test calculations presented to demonstrate the properties of the PIC method as applied to suspension flows include studies of an impulsively accelerated solid boundary moving with constant velocity into a suspension initially at rest, and of suspension flows in shock tubes. Shock tube calculations are also used to demonstrate the need for a fine mesh in order to resolve details of the relaxation zone behind the shock, and to investigate instabilities arising from too large a time increment.

aco ¢

cp , e u

Ca c~ d E f F h I K 1 L m

n

N, Nu P Pr Q r

Re t T

N O M E N C L A T U R E

artificial viscosity coefficient speed of sound specific heats of gas specific heat of particulate phase material drag coefficient particle diameter total energy drag force on a single sphere force per unit volume of mixture exerted by particles on fluid coefficient of heat transfer between particles and fluid specific internal energy fluid thermal conductivity interparticle spacing characteristic length of flow field mass associated with an individual mass point number of particles per unit volume of mixture number of mass points in a cell Nusselt number fluid pressure fluid Prandtl number heat transfer rate per unit volume of mixture from particle to fluid particle radius particle Reynolds number time temperature

I l l

112 ANDREW S. LEVINE a n d BERNARD OTTERMAN

U ~ k

X

7

At Ax E

tx lJ

P (7

O'p

'7 s

Tt ~ 'Tv

velocity effective velocity of a mass point Eulerian coordinate specific heat ratio of gas particle-fluid specific heat ratio time increment cell length particulate phase volume fraction particulate phase mass loading ratio fluid molecular mean free path fluid dynamic viscosity fluid kinematic viscosity mixture phase density (mass of material per unit volume of mixture) density of fluid phase density of particulate phase material time for shock to reach a particle characteristic times associated with thermal and velocity equilibration of a particle

Subscripts e equilibrium E equivalent gas f frozen h shock-heated region i cell j index in Cartesian tensor notation k mass point o pure gas; undisturbed state p particulate phase r rarefied region

Superscripts n time increment ~ property determined in Stage One of Particle-In-Cell calculation

- time-averaged property used in Stage One of Particle-in-Cell calculation

1. I N T R O D U C T I O N

THE HIGH speed flow of gas containing solid particles or liquid droplets often involves situations in which the phases are not in dynamic or thermal equilibrium. Examples include the motion of clouds of dust or droplets induced by explosions or by flight-vehicle aero- dynamics, and the injection of a dispersed phase into a high velocity gas stream. Such problems generally will not admit of a closed form solution, but lend themselves to numerical solution techniques. However, the method selected must be applicable to regions of the flow field in which no particulate material is present, while matching the suspension and pure gas regions. In addition, the calculation method should make it possible to follow individual particles. The ability to include particle reflections and collisions is also desirable.

The Particle-In-Cell (PIC) method[I ,2] is a calculation technique which encompasses the aforementioned features. This paper presents the results of test calculations illustrating the use of a modified form of the PIC method for suspension flows. Included are studies of an impulsively accelerated solid boundary moving at a constant velocity into a suspension initially at rest, and of the behavior o f suspension flows in shock tubes. The numerical properties of the calculation method also are considered.

It will be recalled that in the P1C calculation the flow field is divided into a grid of small regions fixed in space, called cells. In the first stage of the calculation the governing equations are interpreted as Lagrangian expressions--finite-difference forms of which are used to determine the new velocities and the new specific internal energies, after one time step, for the material of each phase initially in a given cell. Interactions between the phases

Analysis of unsteady supersonic two-phase flows by the particle-in-cell method 113

of a gas-particle flow are calculated in Stage One using cell quantities from the previous time step. Stage Two deals with the transport of material among cells during the time step. The material of each phase is represented as discrete masses, typically between two and twenty to a cell. These masses are customarily referred to as 'particles' in gas dynamic calculations; however, they are termed 'mass points' herein to avoid confusion with the particles of the suspension.

The distance each mass point moves during the material transport is the product of the time step size and the ' effective velocity' of the mass point. The latter is a weighted average of the velocity of the cell occupied by the mass point before the transport, and the velocity of the adjacent cell. The weighting factor for the adjacent cell velocity is proportional to the proximity of the mass point to that cell. It should be noted that the location of each mass point is stored and continually updated during the entire calculation.

The portion of Stage Two following the material transport is termed 'repartitioning.' It can be interpreted as a procedure which establishes the material systems contained within each cell at the end of the time step, thereby providing the properties of each phase for the next Stage One calculation. The modified calculation scheme is summarized in the Appendix and described in detail in [3].

The Particle-ln-Cell procedure does not require that the particulate phase be considered a continuum; in fact, the behavior of a single particle can be obtained if it is represented by a mass point. However, when a large number of mass points is used to represent the particulate phase, the averaging procedure used in the Stage Two calculation of the new cell properties corresponds to the commonly assumed 'particle continuum.'

2. A N A L Y S I S

Governing equations The kagrangian differential equations of momentum and energy for the fluid velocity, u,

and specific internal energy, I, are:

duj c~p P d-T --- - d--~j. + r j (1)

dI t~uj P -dr = - Ox'-'-~j + Q + Fg(uej - uj). (2)

Equations (1) and (2) apply to a flow in which viscous and heat-conduction effects are negligible.

The viscous stresses which the particles exert on the gas are included implicitly in the determination of Fj, the drag force per unit volume, while heat interactions between the phases are included in the evaluation of Q, the heat transfer rate per unit volume.

The equations governing the particulate phase are:

due~ - F j (3) PP d-S- =

PP d--~ = - Q (4)

In equation (3), it has been assumed that the force on the particles due to the pressure gradient in the fluid can be neglected. It can be shown that the pressure force on the particles per unit volume of mixture is given by (pp/trp)@/~xj. Since, in general pp/tr~, ,~ 1, this

CAF Vol. 3 No. I--H

114 ANDREW S. LEVINE and BERNARD OTTERMAN

assumption is valid. It has also been assumed that it is possible to neglect the contribution of the particulate phase to the pressure of the suspension. Rudinger[4] found that the pressure contribution of the particulate phase need be considered only at the extremely large particle number densities which would correspond to the simultaneous existence of mass fractions on the order of unity, and particle diameters on the order of 10 -2/~m. Since the present work deals with particles whose diameters are between about 1 and 100/~m, the particulate phase pressure contribution can be neglected. Furthermore, parameters typical of supersonic gas-particle suspension flows lead to interparticle spacings which are large compared to the particle diameter as shown by Liu[5]. Therefore, collisions among particles can be neglected, and the motion of the particulate phase is influenced by the drag force exerted on the particles by the gas and by force field interactions. The latter, including gravity, are neglected in this study.

The mixture phase densities p and pp are related to the material densities p and trp through the particulate phase volume fraction, e:

p = a(l - e), (5)

pp = ape. (6)

The set of equations is completed by the equation of state of the gas phase:

p =f (a , I). (7)

In order to determine the variation of the quantities uj, upj, I, 1,, p, p, p, , a and e with position and time, the drag force, Fj, and heat transfer, Q, between the phases must be expressed in terms of these quantities. The manner in which this is accomplished is described in the following paragraph.

Phase interactions

Regardless of the mathematical model used to represent the suspension flow, its applic- ability obviously depends on the correct representation of the laws governing the exchange of momentum and energy between the phases. The approach used herein is to first calculate the drag force and heat transfer for a single spherical particle moving as if in a continuum of infinite extent. For this approach to be valid, the particle diameter, d, must be large compared to the fluid molecular mean free path, 2, and furthermore, d <~ 1. These require- ments are satisfied by the suspension flows considered herein, as demonstrated by Liu[5]. The single particle drag force and heat transfer are then multiplied by the particle number density per unit volume to give the magnitude of the corresponding phase interaction per unit volume of mixture.

Theoretical and experimental determinations of the drag on a sphere moving at constant velocity relative to an infinite, isothermal, incompressible medium are represented by a widely accepted ' standard drag curve' which is a plot of the drag coefficient, Co, vs the particle Reynolds number, Re. An expression given in [6] which matches the standard drag curve is:

C o = 0.48 + 28(Re) -°'85. (8)

In the limit as the Reynolds number approaches zero, equation (8) approximates Stokes drag law; i.e.,

Co = 24 Re-1. (9)


Although it is widely recognized that factors such as compressibility, fluid turbulence, particle acceleration and shape, and particle-fluid temperature difference lead to variations from the standard drag curve, the present state of knowledge is not sufficient to make possible the accurate prediction of these effects. Therefore, equation (8), or the limiting condition represented by equation (9), are used herein.

Heat transfer effects are represented by formulas which give the dependence of the Nusselt number, Nu, on the particle Reynolds number and on the Prandtl number, Pr, of the fluid. The limiting value of the Nusselt number of a sphere as the Reynolds number approaches zero is 2.0. For a moving sphere, Knudsen and Katz[7] present an equation based on experimental results:

Nu = 2 + 0.6 Re 1/2 Pr 1/3. (10)

It is to be expected that many of the factors affecting the particle drag will also affect the heat transfer. While existing experimental and theoretical studies bear out this prediction it is generally found that heat transfer is less sensitive to such factors than is the drag force. The calculations which follow employ either equation (10) or the limiting value Nu = 2.0. Shock tube studies in which the phase interaction laws were varied systematically are reported by Otterman and Levine[8].

Treatment of boundaries

Four types of boundaries are considered in the present calculations: inflow, outflow, motionless solid and moving solid, The treatment of the gas phase in the first three cases is as described in [2]. For a solid boundary moving relative to the cells, the calculations are as follows. To obtain the pressure of the cell in which the moving boundary is located, the actual cell volume (which changes as the boundary crosses the cell) is used. The pressure at the moving boundary is set equal to the cell pressure, while the boundary velocity, ui-1, is set equal to the velocity of the moving boundary. In Stage Two of the calculation, the boundary position is updated. If, after the mass points have been moved to their new locations, the boundary has overtaken a gas phase mass point, that mass point is reflected from the boundary. (Gas phase mass points adjacent to a moving boundary acquire the boundary velocity in a small number of time steps; consequently, few such reflections are actually necessary.)

The treatment of the particulate phase at inflow and outflow boundary cells is the same as that of the gas, except that pressure effects are not included. However, the particulate phase is assumed to stick to a solid boundary upon colliding with it; therefore, each particulate phase mass point undergoing such a collision is removed from the system. Were particulate phase mass points to be reflected, the calculation procedure used herein would result in their velocities being averaged together with the velocities of unreflected particulate mass points in the subsequent calculations. Although there exists a physical counterpart to the averaging, namely, collisions among the particles of the suspension, interactions of this type are being neglected herein.

3. TEST CALCULATIONS

The calculations described below involve the one-dimensional propagation of shock and rarefaction waves. It is possible to predict some of these results without resorting to the PIC procedure; consequently, the PIC results can be verified. This other prediction method


is based on the fact that suspension flow properties in the 'equilibrium region' far downstream of a shock can be related to the properties ahead of the shock using the normal shock equations[9], where the equilibrium suspension is considered equivalent to an ideal gas, the properties of which (specific heat, density, etc.) can be defined in terms of the equilibrium parameters of the suspension[4]. For example, the equivalent gas specific heat ratio, Ve, and sound speed, c~, are given by:

",~e =~o( 1 Af-~)( l "~-~'o~) -1, (11)

ce = co(l + r/6)1/2[(1 + 0)(1 + yoq6)] -1/2 (12)

where c o is the sound speed in the pure gas. Thus, the downstream equilibrium suspension flow properties depend only on the upstream properties and the shock speed and are inde- pendent of the nature of the phase interaction--just as the properties on either side of a shock in a pure gas can be related without reference to the nonequilibrium processes occur- ring within that shock.

In unsteady suspension flow calculations, it is convenient to define 'equilibration times' which characterize the times required for the phases to attain velocity or thermal equilibrium. The velocity equilibration time,

2 ffpr 2 rv -- (13)

9 l t

is the time required for a particle to reduce its velocity relative to the gas by e-1 of its original value when Stokes law is used to compute the drag force. Similarly, the thermal equilibration time,

1 ffp r2Cs z~ - (14)

3 k

is the time required for the temperature difference between a particle and the gas to be reduced to e-~ of its initial value when a Nusselt number of 2"0 is used to determine the heat transfer coefficient. Such characteristic times are useful in comparing the times required for the relaxation processes with the time during which the flow is to be studied.

Impulsive boundary mot ion

The problem of an impulsively accelerated boundary piston moving with constant velocity into an initially motionless suspension provides a comparison of the P I C calculation with closed-form analytical results. At problem times which are small compared with the equilibration times, the shock propagates as though no solids are present; the shock speed and flow properties in this case are termed ' frozen '. At large times, there is a region of increasing length ahead of the piston in which the suspension moves at the velocity of the piston; the 'equilibrium' shock velocity at these later times must satisfy the normal shock equations as discussed above. Consequently, the shock speed decreases approaching its equilibrium value as the shock propagates into the suspension. Figure la is a schematic ' x - t ' diagram illustrating the variation of piston and shock positions with time. The change in slope of the shock path represents the decrease in shock velocity with increasing time.

Figure lb shows the manner in which the gas velocity, u, and particulate phase velocity, up, vary with position at the early time, tl, indicated in Fig. la. The distance between the piston and the shock is too short for the particulate phase to achieve velocity equilibrium with the gas; therefore, at small times particles are overtaken by the piston.

Analysis of unsteady supersonic two-phase flows by the particle-in-ceU method 117

t Piston ~

t j - . . . . . J -

I I o. Position-time trajectories piston and shock I I I

u l I I I

Piston ------ Gos velocity velocity / ' ~ . I ~ P°rticul°te vel°cit I

/ j shoc, . = I

b. Velocity distribution ot time T= I 1

u I I

Piston I Gos veloci?y velocity ~ 1

Particulate Shock

D x c. Vel0city distribution at time t 2

Fig. I. Shock wave and velocity profiles produced by an impulsively accelerated piston moving with constant velocity into a gas-particle suspension which is initially at rest, schematic.

The region of equilibrium between the relaxation zone and the piston at a later time, t2, is shown in Fig. lc. Once this condition has been reached, the shock propagates at a constant velocity.

The particular example considered herein concerns a suspension of ten micron glass beads which are assumed to stick to the boundary upon colliding with it. The particulate mass fraction is 30 per cent, and the parameters of the suspension give velocity and temperature relaxation times of 0.66 msec and 0-77 msec, respectively.

Figure 2 shows profiles at 0.8 msec as calculated by the PIC method ( A x = 0.02 ft; At = 0.002 msec). At the problem time shown the temperatures and particulate velocity have not yet acquired their equilibrium values. (The velocity of the gas must at all times be equal to that of the adjacent solid boundary.) The uniformity of the gas properties between the shock and the boundary is due to the relatively small drag given by Stokes drag law, equation (9).

The behavior of several representative mass points, determined by the Particle-In-CeU calculation, is shown in Fig. 3. The locations of the particulate phase mass points coincide with the position-time trajectories predicted by assuming that the particle, once the shock encounters and moves past it, is subjected to the uniform gas velocity, ue, behind the shock (ue is equal to the boundary velocity). Using Stokes law to calculate the drag force gives the particle velocity:

up = 0 t < z~ (15a)

up = ue[1 - e -u-~)#°] t >_ z~ (15b)

118 ANDREW S. LEVINE a n d BERNARD OTTERMAN

'o'_o x

7.0 - - ~ 1400

Tf - - - -

6 . 0 - "re ~ Tp ~ 1200

5-0 I000

4 . 0 - Pe P _ p, coo

3"0 600

Ue

2.0 400

1.0 200

0 0.4 0'6 0,8 1.0 1.2

P o s i t i o n , f f

g tn

4 -

+. "8 o >

Fig. 2. Variation of pressure, velocity, and temperature produced by an impulsively accelerated solid boundary moving with constant velocity into a suspension initially at rest, according to the Particle-In-Cell method. Stokes drag law and Nusselt number of 2.0 are used. Boundary

velocity ~ 496 ft/sec, t = 0.8 msec.

060 E

E i : 040

1.0

0 8O Mass p o i n t s

pot ficulote phose x Gas phase

0.20

_ Equation 05)

OqO

Boundary... - -

O 20 1.30 0-40 050

Posit ion, f t

Fig. 3. Fluid and particulate phase trajectories for the example of Fig. 2.


where z~ is the time required, measured from the start of the calculation, for the shock to reach the particle. The gas phase mass points follow trajectories predicted from one dimensional gas-dynamic formulas[10] for a gas subjected to a shock moving at the frozen shock velocity.

The close agreement between the velocity calculated by the PIC method and that predicted by equation (15) demonstrates that, although the shock is represented by a region of steep property gradients rather than by a discontinuity, this does not lead to errors in the motion of the particulate phase. The foregoing is true provided the time each particle spends within that region is much less than the velocity equilibration time of the particle.

Effects of cell size and time increment

The effects of cell size and time increment on the PIC calculation of a suspension flow were investigated by comparing several shock tube studies, all using a pure gas driver, a mass loading ratio of unity in the driven section, and a pressure ratio of 5 : 1. Comparison with analytical results (e.g., ref.[ll]) is afforded by the fact that the flow of gas particle suspensions in shock tubes at problem times which are large compared to the equilibration times is characterized by the presence of a zone behind the shock in which the phases are in velocity and thermal equilibrium, and by a constant shock speed. Therefore, the PIC results for the equilibrium zone can be compared with those calculated by applying the shock tube equations to an ' equivalent gas' defined in terms of the properties of the suspension as discussed above.

Figure 4 gives the results of a calculation in which a cell size of 0.1 ft was used. The flow parameters at 2.4 msec calculated by the PIC method are compared with the predictions of the shock tube equations. The material density ratio, tr/ap, is sufficiently small (on the order of 10-3) at the driver section pressure of 5 atm to justify neglecting the particle volume

12.0 ,600

Porticle in cell method |

" - - ' S h o c k tube equof ions

to.o ~, ! - - 500 ~J

b ~, =_o × e.o ~o0

~ . , j

6"0 - 300

~- . . o ~oo

2.0 ~00

I I I I 1.0 2-0 3.0 40 5.0 6.0

P o s i t i o n , f t

Fig. 4. Pressure, velocity and temperature profiles for a shock tube with the low pressure end containing a suspension of 10 p.m glass beads in air, according to the PIC method and the shock tube equations. Initial condit ions: pressure ratio = 5.0, co = 1000 ft/sec,

7 /= 1.0, y = 1.4, t~ = 0"5, t = 2.4 msec.


fraction, e, in the shock tube equations using the 'equivalent gas.' The values q = 1-0, and ? = 1.4 used in (11) and (12) give equivalent gas properties 7E = 1.235 and cE = 665 ft/sec for the driven gas. Figure 4 shows that the suspension flow parameters are computed correctly by the Particle-In-Cell method. For this example, the computation time was 24 sec on a CDC-6700.

Property profiles at 0-8 msec for a calculation with cell length 0.1 ft and time increment 0"01 msec are shown in Fig. 5, along with the results obtained with a cell length of 0.02 ft. (A time increment of 0.002 msec was used in the latter case so that the stability parameter c A t l A x would be unchanged.) The five-fold increase in the number of cells per unit distance enhances the resolution of the shock and contact discontinuities and of the rarefaction wave. Such resolution is also necessary for the accurate portrayal of the property profiles in the relaxation zone.

Figure 5 also indicates that a change in Ax produces a qualitative difference in the temperature profiles in the shock heated region. This can be explained by noting that when the cell length is 0.1 ft, the shock heated region encompasses only about five cells at a problem time of 0.8 msec. However, the shock and contact discontinuities occupy about ten cells each[3]; consequently, the regions occupied by the discontinuities overlap when the cell size is large and render the temperatures in the shock-heated region incorrect,

The equilibrium values of pressure, temperature and velocity have been indicated in Fig. 5. There exist two equilibrium temperatures: Te, h corresponding to the temperature in the shock-heated zone, and T~,, corresponding to the suspension through which the rarefaction wave has propagated. The gas temperature in the shock-heated region and the gas velocity between the shock and rarefaction waves are both above their equilibrium values since the gas properties initially assume the ' f rozen ' values u I , p i , TI, h, and TI, r corresponding to a flow with no particulate phase present.

The effect of time-increment size on the P I C calculation was examined in a series of shock tube studies using time increments between 0.001 msec and 0.01 msec. (The cell

12.0 600 p - - ~x=0.02f t

~- \ \ l~x=O.I f t

I0.0 500

, o o o

T , , , _ ~ /

~; 6.0 \ L \ 300 2

> X. 4.0 zoo

2.0 I00

0 . ~ W / / / I l I \klk~l\\\" 0 t.o i.5 2-0 2.s 3.0 3.5 4.0

Position, f f

Fig. 5. Effect o f cell size on pressure, veloci ty and temperature prof i les fo r a shock tube wi th a suspension in the low pressure end only. t = 8.0 >: 10 -4 sec. A l l o ther condi t ions are the same

as for Fig. 4.


12.0 600 i P

I00 "- ~ ~ 500

o,o_ _ y,-~ a.o 400

p,

-o- / l l,oo, o u I i I I o

I.o 1.5 2.0 z.5 3.0 3.5 4.0

Posit ion, f t

Fig. 6. Pressure, velocity and temperature profiles for a shock tube with a suspension in the low pressure end only. At=8.0 x 10-6see. All other conditions are the same as

for Fig. 5.

length was 0.02 ft in each case.) There is no significant difference among the calculations using time steps of 0.004 msec or smaller. The calculation using a time step of 0.008 msec was unstable, as evidenced by the gas-phase property profiles of Fig. 6, although the stability condition that the fluid move less than one cell per time step (umax At < Ax) and the CFL condition (c At < Ax) are both satisfied in this case. Additional computational experiments indicate that the instability is associated with the gaseous phase. A possible interpretation of the condition Umax At < Ax for the PIC method is that the distance a mass point moves in a single time step be less than the spacing between mass points. Then ui At < Ax/Ni; where N~ denotes the number of fluid mass points in cell i. Single-phase PIC calculationsl3] showed that this condition is sufficient for stability. The addition of a particulate phase will in general cause a decrease in both u, and the equivalent sound speed, c~. Consequently, this stability condition will remain satisfied, and is sufficient, for a PIC suspension-flow calculation.

5. CONCLUDING REMARKS

The Particle-In-Cell method has been used to analyze the unsteady flow of gas-particle suspensions. The material comprising each of the phases of the suspension is represented by mass points, while the flow field is divided into small regions fixed in space, called cells. Interactions between the phases are incorporated into the Lagrangian ' Stage One ' equations. A feature of the PIC method which was shown to be particularly useful for the flow analysis of gas-particle suspensions is that the phases are continuously traced as they move through the flow field ; therefore, the path lines of material elements are a straightforward result of the calculation. Another feature of the calculation is that it can be readily applied to situations where the particulate phase does not occupy the entire flow field. Finally, the calculation method is not based on a continuum treatment of the particulate phase; consequently, the motion of individual particles can be studied.

122 ANDREW S. LEVINE and BERNARD O'VrERMAN

A test calculation for an impulsively accelerated solid boundary moving into a suspension initially at rest was presented. Al though a shock wave in a gas is represented in P 1 C

calculations by a region of steep property gradients rather than by a discontinuity, this calculation shows that the mathematical existence of a finite shock width does not lead to significant errors in the motion of the particulate phase.

The calculation procedure described herein was also applied to the investigation of suspension flows in shock tubes and the results were in agreement with the predictions of

the shock tube equations. Shock tube calculations were also used to demonstrate the need for a fine mesh in order to resolve details of the relaxation zone behind the shock. Finally, an investigation of instabilities arising from too large a time increment led to the stability condit ion u i At < A x / N i , where N i denotes the number of fluid mass points in cell i.

REFERENCES

1. Harlow F. H., The Particle-ln-Cell method for numerical solution of problems in fluid dynamics, Proc. Syrup. Appl. Math. 15, 269 (1962).

2. Amsden A. A., The particle-in-cell method for calculation of the dynamics of compressible fluids, Los Alamos Scientific Laboratory Report No. LA-3466 (1966).

3. Levine A. S., A theoretical analysis of unsteady, supersonic, gas-particle suspension flows using the particle in ceil method, Ph.D. Thesis, Northeastern University, Boston, Massachusetts, 1971.

4. Rudinger G., Some effects of finite particle volume on the dynamics of gas-particle mixtures, A I A A J. 5, 1217 (1965).

5. Liu J. T. C., Problems in particle-fluid mechanics, Ph. D. Thesis, California Institute of Technology, Pasadena, California, 1964.

6. Gilbert M., Davis L. and Altman D., Velocity lag of particles in linearly accelerated combustion gases, Jet Prop. 25, 26 (1955).

7. Knudsen J. G. and Katz D. L., Fluid Mechanics and Heat Tramfer. McGraw-Hill, New York (1958). 8. Otterman B. and Levine A. S., Analysis of gas-solid particle flows in shock tubes, A1AA J. 12, 579

(I 974). 9. Rudinger G. and Chang A., Analysis of nonsteady two-phase flow, Phys. Fluids 7, 1747 (1964).

10. Shapiro A. H., The Dynamics and Thermodynamics o f Compressible Fluid Flow. Ronald, New York (1953.)

11. Gaydon A. G. and Hurle 1. R., The Shock Tube in High Temperature ChemicalPhysics. Reinhold, New York (1963).

APPENDIX

PARTICLE- IN-CELL CALCULATION PROCEDURE FOR TWO-PHASE FLOWS

Difference equations (stage otle) To obtain the pressure P7 the material density of the gas, a, is calculated from equations (5) and (6), and

then used with the ideal gas equation of state to give:

P7 = ,oT('Z- 1)/~[I -- P--~Pf]-' (A1) I. fit, J

where superscript n denotes the time-step number. The Stage One equations are:

At (Pi-i/2 + q~-t/2) --P~+1/2 + ql+x/2) + F~' (A2) ~7 : u~' - p7 Ax

Or7 up7 -- F~ :-kt/pt,7 (A3)

~ t : n - i7 = 17 + p--f~ l[--pi(ut+a/2 -- t-ti-,/2) ÷ ul(qi-1/z --q,+i/z)

- - (qi-l/: ul-1/2 -- qi+l/2 u~+t/2)] (I/Ax) + Fr(ft,z -- t~l) + QT} (A4)

L7 = lt,7 -- Q7 At/pt,7. (A5)


The terms appearing in equations (A2) through (AS) are defined as follows:

n Pl± 1/2 ~ (PT±, + p7)/2

+++ ~ (u] + t77)/2

F7 = (3/16)Co Re(u.7 -- uT)ppTv/(apr 2)

Q7 = (3/2)(NuT~7 -- 77)p.7 K/(~. r2).

(A6)

(A7)

(A8)

(A9)

When the particulate phase consists of a single particle, (A8) and (A9) become identical with expressions for the drag and heat transfer of a single particle.

The quantity qT± 112 is an 'artificial viscosity' term which is added to dampen cell-property fluctuations which result from the instability of the equations, and is calculated as in[2]:

NpNP±tm [ ~ ( u 7 - - u?± t)] . q71±112 = aco (N? + NT±I)(2Ax)

(AIO)

Transport and repartitionin9 (stage two)

The momentum of the new system corresponding to each phase which occupies a cell is divided by the mass of that phase in the cell to obtain the new cell velocity u7+1:

uT+t - 1 [ ~ (mt~)] (Al l ) rnNp +x L k = l /

where the summation is taken over the Ns fluid-phase mass points which now occupy cell i, and t~, denotes the Stage One velocity of the donor cell (in which mass point k was located prior to the material transport). Similarly, u N+' is given by:

+ ' ] mp N ~ ' ' ' - - ~ , (ms ~p~) i (A 12)

where Np, denotes the number of mass points which represent the particulate phase in cell i. The new specific internal energy of each phase is obtained by subtracting the corresponding specific

kinetic energy, ½(ugh") 2 or ½(u~+') 2, from the new value of the total energy per unit mass. This procedure gives:

Nl

1~,+ +. - mN7 + ' I [5,= m[T~' + ½(ff~)']],-- ½(uT+') 2 (AI3)

and

, ] m. N~, + - - - - i mz[[P~' ~- ½(/~P~)2I - - ~ ( U g / + l ) 2 ' (A14) 1 l

In equation (A13), i~ denotes the Stage One specific internal energy of the donor cell associated with fluid mass point k. Similarly, lr~, in equation (AI4) is associated with donor cell corresponding to particulate phase mass point k. Equation (A14) produces the averaging of particulate phase properties over the cell, which corresponds to the assumption of a ' particle continuum'.

analysis of unsteady supersonic two-phase flows by the particle-in-cell method

Documents