Transient phenomena in one-dimensional compressible gas–particle flows

Shock Waves (2009) 19:67–81DOI 10.1007/s00193-009-0190-1

ORIGINAL ARTICLE

Transient phenomena in one-dimensional compressiblegas–particle flows

Y. Ling · A. Haselbacher · S. Balachandar

Received: 19 July 2008 / Revised: 17 December 2008 / Accepted: 2 February 2009 / Published online: 26 February 2009© Springer-Verlag 2009

Abstract The transient behavior of compressible gas–particle flows produced in shock tubes with particle-ladendriver section is studied. Particular attention is focused onthe time scales with which the solution approaches the equi-librium state. Theoretical estimates indicate that the gas andparticle contact surfaces equilibrate first, followed by theshock wave, and finally by the expansion fan. The estimatesare in good agreement with numerical simulations. The sim-ulations also show that the approach to equilibrium conditionof the shock speed is non-monotonic (monotonic) if the massfraction of particles initially located in the driver section isbelow (above) a particle-diameter dependent critical value.For the speed of the particle contact surface, the reverse trendsare observed.

Keywords Gas–particle flow · Unsteady flow · Shock tube

PACS 47.55.Kf · 47.37.+Q · 47.11.Df

Communicated by E. Timofeev.

Y. LingDepartment of Mechanical and Aerospace Engineering,University of Florida, MAE-B 237, P. O. Box 116300,Gainesville, FL 32611-6300, USAe-mail: [email protected]

A. Haselbacher (B)Department of Mechanical and Aerospace Engineering,University of Florida, MAE-B 222, P. O. Box 116300,Gainesville, FL 32611-6300, USAe-mail: [email protected]

S. BalachandarDepartment of Mechanical and Aerospace Engineering,University of Florida, MAE-A 231, P. O. Box 116300,Gainesville, FL 32611-6300, USAe-mail: [email protected]

1 Introduction

Compressible particle-laden flows are encountered in manyfields, such as solid-propellant rocket motors and detonationof multiphase explosives. The presence of inert particles cansignificantly affect the flow through momentum and energyexchanges. These exchanges can introduce interesting newphysical phenomena that are not present in a single-phasecompressible flow. The classic shock-tube problem, see, e.g.,[22], can be employed to study compressible gas–particleflows at a fundamental level.

The shock-tube problem is attractive for several reasons.First, an analytical solution is available for the single-phasecase. Second, the shock-tube problem includes the basic com-pressible flow features such as an unsteady expansion fan, acontact discontinuity, and a shock wave. Third, the numericalsolution of the shock-tube problem is inexpensive, makingsystematic parametric studies feasible. Past studies of themultiphase shock-tube problem can be divided into threetypes depending on whether particles are located only in thedriven section, only in the driver section, or in both sec-tions. Studies of the first type mainly focus on the interactionbetween the particles and the shock wave, while studies ofthe second type focus on the interaction between the particlesand the expansion wave.

The majority of past studies of gas–particle flow in theshock tube have placed the particles in the driven sectionand investigated shock-particle interaction, see, e.g., Carrier[2], Soo [21], Kriebel [12], Rudinger [17], Miura and Glass[15], Sommerfeld [20], and Saito [19]. This setup is rele-vant to the study of the propagation of shock waves in adusty environment such as the atmosphere. Because of thefinite mechanical and thermal inertia of particles, the particlescannot adjust immediately to the instantaneous changes ofvelocity and temperature through the shock wave. As a result,

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68 Y. Ling et al.

the interaction between a shock wave and particles leads tothe so-called frozen, relaxation, and equilibrium regions thatare discussed in more detail below. Carrier [2], Soo [21],Kriebel [12], and Rudinger [17] studied the relaxation regionnumerically.

The case of particles in the driver section does not appearto have been studied intensively despite its importance inapplications such as volcanic eruptions and particulate explo-sions. Two notable efforts are by Rudinger and Chang [18]and Elperin et al. [4]. In general, the interaction between anexpansion wave and particles is far more complex than thatbetween a shock wave and particles. Frozen and equilibriumsolutions can also be defined, but the approach of the solu-tion to equilibrium is far slower than for the interaction witha shock wave.

In the present work, we investigate the problem of a shock-tube with particles in the driver section. The influence of theparticle phase is investigated in detail through parameterssuch as mass loading, mechanical, and thermal time scaleson the propagation of the shock, gas and particle contact sur-faces, and the expansion fan. Particular attention is paid to thetime scale on which the equilibrium solution is approached.A simple analysis is presented to estimate the time scale onwhich the gas and particle constants, shock wave and expan-sion fan approach their equilibrium states. These estimatesare compared against the corresponding computed values.

2 Theory

2.1 Assumptions

We make the following assumptions: (1) The fluid may berepresented by a perfect gas. (2) The fluid motion may beregarded as inviscid, so the fluid viscosity and conductivityare neglected except in the interaction with the particles. (3)The particles are inert, rigid, and spherical. (4) The particleshave constant heat capacity and uniform temperature distri-bution. (5) The volume fraction of the particles is negligible.(6) The only force acting on the particles is the viscous dragforce. (7) The particles do not collide with each other.

2.2 Governing equations

The equations governing the gas phase are convenientlyexpressed in the Eulerian frame of reference. The particlephase can be treated using either the Lagrangian approach,where each individual particle is tracked, or using the Eule-rian approach, where the particle phase is treated as a con-tinuum. We have computed solutions using both approachesand obtained results that compared very well. For brevity, wepresent only the results obtained using the Eulerian methodfor the particle phase.

In the Eulerian–Eulerian framework, the mass, momen-tum and energy equations of the gas phase can be writtenas

∂(ρY g)

∂t+ ∂(ρY gug)

∂x= 0, (1)

∂(ρY gug)

∂t+ ∂[ρY g(ug)2]

∂x+ ∂pg

∂x= − f p, (2)

∂(ρY g Eg)

∂t+ ∂(ρY g H gug)

∂x= −u p f p − q p, (3)

where ρ, Y, u, p, E , and H represent the density, massfraction, velocity, pressure, total energy, and total enthalpy.The superscripts g and p indicate the gas and particle phases,respectively. Quantities without superscript denote mixturevariables. The total enthalpy of the gas H g is given by

H g = Eg + pg

ρY g. (4)

The corresponding mass, momentum, and energy equa-tions of the particle phase are:

∂(ρY p)

∂t+ ∂(ρY pu p)

∂x= 0, (5)

∂(ρY pu p)

∂t+ ∂[ρY p(u p)2]

∂x= f p, (6)

∂(ρY p E p)

∂t+ ∂(ρY p E pu p)

∂x= u p f p + q p. (7)

The terms f p and q p that appear on the right-hand side ofthe equations represent the viscous drag force on the parti-cles and heat transferred to the particles per unit volume dueto the momentum and energy exchanges between the phases.The expressions for f p and q p are

f p = ρY p ug − u p

τ p

CD Re

24, (8)

and

q p = ρY pC p T g − T p

τpθ

Nu

2, (9)

where C p is the specific heat of the particles, and τ p and τpθ

represent the mechanical and thermal particle response timesgiven by

τ p = ρ p(d p)2

18µg, (10)

and

τpθ = C pρ p(d p)2

12κg, (11)

where d p is the particle diameter and µg and κg are thedynamic viscosity and thermal conductivity of the surround-ing gas.

In (8) and (9), CD , Re, and Nu are the particle drag coeffi-cient, Reynolds number, and Nusselt number. The Reynolds

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Transient phenomena in gas-particle flows 69

number is defined in terms of relative velocity between aparticle and the surrounding gas as

Re = ρg|ug − u p|d p

µg. (12)

In the present work, we choose the drag coefficient andNusselt number to be given by the following correlations,see [3] and [23],

CD = 24

Re(1 + 0.15Re0.687) + 0.42

(1 + 42500

Re1.16

)−1

,

(13)

and

Nu = 2 + (0.4Re0.5 + 0.06Re2/3)Pr0.4 , (14)

where Pr is the Prandtl number,

Pr = µgCgp

κg, (15)

and Cgp represents specific heat at constant pressure of the gas.

With the definition of the Prandtl number, the ratio of particlemechanical and thermal response times can be expressed as

τ p

τpθ

= 2

3

1

Prδ, (16)

where δ = C p/Cgp is the specific heat ratio between the

particles and the gas.Assuming that the gas follows the perfect gas law and the

particles not to contribute to the pressure, the equation ofstate can be written as

pg = ρY g RgT g . (17)

2.3 Frozen and equilibrium solutions

Consider the problem of a shock tube with particles in thedriver section. Figure 1 shows a typical x − t diagram, wherethe numbers 4, 3, 2, and 1 represent the regions to theleft side of the expansion fan, between the expansion fan andthe gas contact surface, between the gas contact surface andthe shock, and to the right side of the shock, respectively. Theshaded area indicates the presence of particles. The waves inFig. 1 are representative of cases with very low mass loading.Appropriate x − t diagrams for larger mass loadings will beshown in the results section.

The driven section is fully characterized by the gaspressure pg

1 , temperature T g1 , gas constant Rg

1 , and ratio ofspecific heats γ

g1 . The driver side is similarly characterized

by the gas pressure pg4 , temperature T g

4 , gas constant Rg4 ,

and ratio of specific heats γg4 , but in addition the suspended

particles are characterized by their diameter d p, density ρ p,number density n p

4 (number of particles per unit volume),and specific heat C p. Assuming a mono-disperse uniform

1

24

t

3

xDiaphragm

C

S

T

H PC

Fig. 1 Schematic x − t diagram showing the region numbering usedin this article, where H , T , PC , C , and S denote the head and tail ofthe expansion fan, the particle contact surface, the gas contact surface,and the shock, respectively; shading indicates the presence of particles

distribution of spherical particles, the volume fraction φp4

and mass fraction Y p4 of particles in the driver section can be

expressed as

φp4 = n p

4π(d p)3

6, (18)

and

Y p4 = ρ pφ

p4

ρ pφp4 + ρ

g4 (1 − φ

p4 )

. (19)

The gas in the driver and driven sections and the particlesare at rest initially (ug

1 = ug4 = u p

4 = 0) and the particlesin the driver section are in thermal equilibrium with the gas(T g

4 = T p4 ). In the present study, we focus on the regime

where the mass fraction of particles is large enough so thatthe back effect of particles on the flow is important. However,we consider the particle density to be large enough so thatthe particle volume fraction is small despite the large massfraction. As a result, particle–particle interactions and othervolumetric effects can be ignored. The governing equationspresented above are consistent with this regime.

Once the shock-tube diaphragm is ruptured, a shock wavepropagates into the driven section, followed by the gas con-tact surface separating the gases originally contained indriven and driver sections. Because of the finite inertia of theparticles, the particle contact surface (the interface betweenparticle-laden gas and pure gas) will lag the gas contact sur-face, as indicated in Fig. 1. Simultaneously, an expansionwave propagates into the driver gas. The presence of theparticles breaks the self-similarity of the pure gas case, andhence precludes an analytical solution. Before presenting ourresults, we first consider some limiting cases as guidelines.

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70 Y. Ling et al.

2.3.1 Frozen limit

Immediately after the rupture of the diaphragm, while thegas responds instantaneously, the particles can be taken tobe stationary. As considered by Rudinger and Chang [18],a useful limit will be to assume that the particles are frozenat their initial velocity and temperature, and to consider thecorresponding gas flow. In this so-called “frozen limit,” thegas flow can be studied assuming that the particle contactsurface remains fixed since the particles are stationary.Accordingly, the particle volume fraction is equal to theinitial volume fraction.

If we further consider the volume fraction of particles tobe negligible, the frozen flow behaves as a pure gas flow, i.e.,as if the driver section contains only the high pressure gaswithout the particles. Even for finite but small volume frac-tion, the frozen flow behaves like pure gas flow immediatelyfollowing the rupture of the diaphragm. For the present study,as the particle volume fraction is kept small, the analyticalpure-gas solution provides an adequate approximation forthe frozen limit. Therefore, we use the pure-gas solution torepresent the frozen limit in this article.

It should also be noted here that near the head, theexpansion wave is moving into stationary particles. Since theexpansion wave leads to continuous velocity changes, the gasvelocity near the head is quite small. Thus the propagation ofthe head of the expansion wave can be considered to followthe pure gas results at all times. In fact, the flow near the headof the expansion wave is governed by the acoustic equation,implying that the head travels into the driver section at thespeed of sound in pure gas, as observed in the computationalresults of Rudinger and Chang [18] and Elperin et al. [4].

The speeds of the shock wave, gas contact surface, andthe head and tail of the expansion fan in this pure gas limitare given by the familiar shock-tube equation, see, e.g., [22],

pg4

pg1

= 2γg1 M2

s, f − (γg1 − 1)

γg1 + 1

×(

1 − ag1

ag4

γg4 − 1

γg1 − 1

M2s, f − 1

Ms, f

)− 2γg4

γg4 −1

, (20)

where Ms, f represents the shock Mach number in the puregas limit. With the given initial conditions and gas proper-ties specified upstream and downstream of the diaphragm,the above equation can be used to obtain the shock Machnumber. The speeds of the waves can then be expressed as

us, f = Ms, f ag1 , (21)

uc, f = ug2 = ug

3 = 2ag1

γg1 + 1

(Ms, f − 1

Ms, f

), (22)

uh, f = −ag4 , (23)

ut, f = ug3 − ag

3 , (24)

where us, f , uc, f , uh, f , and ut, f represent the speeds of theshock, contact surface, head, and tail of the expansion fan,respectively. The subscript f indicates the pure gas limit.

2.3.2 Equilibrium limit

At very late times, the particles (except for a small regionclose to the head of the expansion fan) can be considered tobe in equilibrium with the surrounding gas, i.e., u p = ug andT p = T g . Under these equilibrium conditions, the mixtureof gas and particles can be considered to behave like a gaswith modified properties (a so-called dusty gas, see Marble[14]). Governing equations for the mixture can be derived byadding those for the gas and the particles. The equilibriumspeeds of the shock wave, gas contact surface, and the headand tail of the expansion fan can then be determined from

pg4

pg1

= 2γ1 M2s,e−(γ1−1)

γ1+1

(1− a1

a4

γ4−1

γ1−1

M2s,e − 1

Ms,e

)− 2γ4γ4−1

,

(25)

where Ms,e represents the equilibrium shock Mach number,and

us,e = Ms,ea1,e, (26)

uc,e = u2,e = u3,e = 2a1,e

γ1 + 1

(Ms,e − 1

Ms,e

), (27)

uh,e = −a4,e, (28)

ut,e = u3,e − a3,e, (29)

where the subscript e indicates the equilibrium limit. Theequilibrium speed of sound a of the mixture is

ae = √γ RT g, (30)

where γ and R represent the specific heat ratio and the gasconstant of the mixture,

γ = γ g1 + Y p

1−Y pC p

Cgp

1 + γ g Y p

1−Y pC p

Cgp

, (31)

R = (1 − Y p)Rg. (32)

Because Ms,e < Ms, f , it can be shown that the expansionfan in the equilibrium limit is always stronger than that inthe pure gas limit. Furthermore, it can be demonstrated thatas Y p

4 increases, the pressure ratio across the expansion fanbecomes stronger while that across the shock decreases.

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2.4 Time scale analysis

As discussed above, immediately following the rupture ofthe diaphragm, the gas–particle flow behaves as a pure gasflow and at large times the gas–particle flow behaves as anequilibrium flow. The change from the initial pure gas flowto the self-similar equilibrium state occurs over a transitionperiod. Here we will examine the time scales that dictate theduration of this transition phase.

Because τ p and τpθ represent the time scales on which the

particles adjust to the ambient flow, for times much largerthan both τ p and τ

pθ , the equilibrium state can be consid-

ered to exist. Since the flow within the expansion fan con-stantly changes with time, perfect equilibrium for particlescan never be achieved. Nevertheless, as time evolves, whenthe expansion region is sufficiently broad, both the tempo-ral and spatial variation of the flow within the fan becomequite small. Since the time scale of the flow continues toincrease with the expansion, the Stokes number of a givenparticle decreases over time to allow an approximate equi-librium description of particles in terms of the local fluidproperties (see Sect. 2.4.4).

2.4.1 Mechanical time scale of particle contact surface

Once the diaphragm is ruptured, the gas contact surfacemoves instantly into the driven section with a speed givenby the pure gas solution. In contrast, the particle contact sur-face initially has zero velocity. Only as the particles that wereinitially at the diaphragm are accelerated by the surroundinggas, does the particle contact surface move into the drivensection. Thus the ambient gas velocity seen by the particlesat the particle contact surface is ug

3, f at t = 0+ and eventually

changes to ug3,e for t � τ p. The time it takes for the particle

contact surface to reach the fluid velocity can be estimatedfrom the particle equation of motion,

du p

dt= ug

3 − u p

τ p

CD Re

24, (33)

where CD , Re, and τ p have been defined in (10), (12), and(13) with ug = ug

3 , and ug3,e ≤ ug

3 ≤ ug3, f . If, for simplicity,

we ignore the time variation of the ambient gas velocity seenby the particle contact surface, use the particle time scale τ p

as the reference time scale, and define t ′ = t/τ p, then (33)can be expressed as

dRe

dt ′= −Re

CD Re

24= −Re(1 + 0.15Re0.687). (34)

In deriving the above relation, we only use the first term ofthe drag law given in (13); this approximation is valid forRe < O(104).

We arbitrarily define the mechanical equilibration timescale t p∗ of the particle contact surface to be the time it takes

log(Re0)

tp*/τ

p

-4 0 4 80

1

2

3

4

5

Fig. 2 Variation of dimensionless mechanical equilibration time scaleof the particle contact surface with Re0

for the velocity difference between the particles at the parti-cle contact surface and the surrounding gas to decrease to 1%of the initial velocity difference. Then (34) can be integratedto obtain

t p∗

τ p= 2.456 ln(100) − 1

0.687ln

⎡⎢⎣ Re1.687

0 + Re00.15(

Re0100

)1.687 + Re015

⎤⎥⎦ ,

(35)

where Re0 is the particle Reynolds number based on theinitial velocity difference between the gas and particles. Inview of the constant velocity approximation for the gasbehind the shock wave made in deriving the above estimate,Re0 can be chosen to be based on the pure gas velocity ug

3, f ,

the equilibrium gas velocity ug3,e, or an intermediate value.

Clearly t p∗ scales as τ p and in the limit of very small parti-cles (i.e., Re0 → 0), we have t p∗/τ p → ln(100) ≈ 4.605.With increasing particle size, as the particle Reynolds numberincreases, the correction to Stokes drag plays an importantrole and the ratio t p∗/τ p decreases, as shown in Fig. 2.Thus, although the particle time scale τ p scales as (d p)2, themechanical equilibration time scale of the particle contactsurface shows a far smaller variation with increasing particlesize.

2.4.2 Thermal time scale of particle contact surface

The thermal time scale of the particle contact surface canbe established in a similar manner. The particles originallyat T p

4 at the particle contact surface initially “see” the gastemperature to be the pure gas temperature T g

3, f and at long

123

72 Y. Ling et al.

log(Re0)

tp* θ/t

p*

-4 -2 0 2 40

5

10

15

20

25

30

35

40

45

50

τp / τpθ=1.0

τp / τpθ=0.58

τp / τpθ=0.29

Fig. 3 Variation of equilibration time scale ratio of the particle contactsurface with Re0

times the equilibrium gas temperature T g3,e. Starting from (7),

we obtain with (8),

dT p

dt= Nu

2

T g − T p

τpθ

. (36)

We introduce θ = T g − T p, use (14), and ignore the timevariation in the gas temperature seen by particle contactsurface, to rewrite the above equation as

1

θ

dθ

dt ′= −τ p

τpθ

[1 + 0.5

(0.4Re0.5 + 0.06Re2/3

)Pr0.4

].

(37)

As before, the thermal equilibration time scale of the parti-cle contact surface t p∗

θ can be defined arbitrarily as the timeit takes for the temperature difference between the particlesat the particle contact surface and the surrounding gas todecrease to 1% of the initial temperature difference. For verysmall particles, Re � 1, Nu → 2, and hence

t p∗θ

τ p= ln(100)

τ p

τpθ

, (38)

where τ p/τpθ is given by (16). For larger particles with finite

Reynolds numbers, (37) can be integrated numerically andthe resulting t p∗

θ /t p∗ is plotted in Fig. 3 as a function of Re0

for different values of τ p/τpθ . Clearly, over a wide range of

Reynolds numbers and mechanical and thermal time scalesof the particles, the equilibration time scales are compara-ble. Together, they characterize the time scale on which theparticle contact surface approaches equilibrium.

In the present problem, the approach to equilibrium isexpected to happen in the following manner. A short time

after the diaphragm is ruptured, the gas velocities and tem-peratures in the expansion fan and between the fan and theshock initially follow their standard evolution as in puregas. As the head of the expansion fan propagates into thedriver section, the particles begin to be accelerated by thesurrounding gas. The velocity of the particle contact sur-face increases steadily, and the temperature of the particlecontact surface decreases similarly. During the approach toequilibrium, owing to finite inertia, the particle velocity istypically smaller than the local fluid velocity and the parti-cle temperature is larger than the local gas temperature. Aswe will see later, under certain circumstances, the particlevelocity can briefly overshoot the local fluid velocity. Theresulting momentum and energy exchange couples back tothe gas, whose velocity and temperature evolve to the equilib-rium state as a result. The time scale on which the particles atthe particle contact surface approach the local gas propertiesis of the same order of magnitude as the time scale on whichthe gas properties at the particle contact surface approach theequilibrium state.

At intermediate times, when the flow is away from equilib-rium, there is strong momentum and energy coupling betweenthe gas and the particles, and as a result both the gas and par-ticle properties show continuous variation from the head ofthe expansion fan to the particle contact surface. In otherwords, while the equilibrium state is approached, the tail ofthe expansion fan cannot be discerned very clearly. Further-more, only at equilibrium will the gas and particle contact sur-faces move at the same speed. Until equilibrium is reached,the gas contact surface moves faster than the particle contactsurface. The integrated effect is that the gas contact surface isa finite distance ahead of the particle contact surface at equi-librium. Nevertheless, it can be argued that since the mech-anisms driving the contact surfaces towards equilibrium arethe same, we can expect the gas contact surface and the par-ticle contact surface to reach equilibrium on the same timescale t∗c , which can be taken to be the larger of t p∗ or t p∗

θ ,i.e.,

t∗c = max(t p∗, t p∗θ ). (39)

2.4.3 Shock time scale

Although the shock wave propagates into pure gas in thepresent problem, and hence does not come into direct con-tact with the particles, the interaction of particles with theexpansion fan creates disturbances that propagate forwardto decelerate the shock wave. As a result, the shock speedstarts out at the pure-gas value and evolves to the equilibriumvalue. The deceleration of the shock wave in turn sends outdisturbances that affect the interaction between the particlesand the expansion fan. Based on this sequence of cause andeffect, it can be anticipated that the time scale on which the

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shock approaches equilibrium, defined arbitrarily as the timeit takes for the shock speed to reach 1% of its equilibriumvalue, is longer than t∗c . The distance between the contactsurface and the shock location increases steadily with time.The position (relative to the location of the diaphragm) atwhich the contact surfaces reach their equilibration can beestimated as (us,e − ug

2,e)t∗c . The speed of propagation of

information from the contact surface to the shock is given bya2,e +ug

2,e −us,e. Thus, the time scale for shock equilibrationcan be crudely estimated as

t∗sτ p

=(

t∗cτ p

+ M2,e

)(γ g + 1)(M2

s,e − 1)

2γ g M2s,e − (γ g + 1)

. (40)

2.4.4 Expansion fan time scales

Although the tail of the expansion fan was estimated toapproach equilibrium together with particle and gas contactsurfaces, the expansion fan in its entirety can be expected tosettle to the equilibrium state on a much slower time scale.Following the arrival of the head of the expansion fan, theparticles have just started to move. Thus it can be argued thatthe head of the expansion fan always moves at the speed ofsound of the pure gas in the driver section. After the expan-sion fan has broadened sufficiently, the velocity variationwithin the fan becomes small enough for particles to followthe gas closely. Again we define equilibrium to imply thatthe particle velocity is within 1% of the local gas velocity.Since the flow within the expansion fan is unsteady, we esti-mate the velocity difference between the particles and thegas using the equilibrium Eulerian approximation of Ferryand Balachandar [5] as

u p − ug ≈ τ p Dug

Dt= τ pug dug

dx, (41)

where u p and ug represent the local velocities of the parti-cles and the gas, and Dug/Dt and dug/dx represent the localmaterial derivative and spatial derivative of the gas velocity,respectively.

As equilibrium is approached, the gas velocity gradientcan be estimated based on the equilibrium gas velocity withinthe expansion fan, where the gas velocity changes from zeroto ug

3 over the fan thickness (a4,e − a3,e + ug3,e)t . Based on

these arguments, the time scale on which the bulk of theexpansion fan approaches equilibrium can be estimated as

t∗eτ p

≈ 100ug

3,e

a4,e − a3,e + ug3,e

. (42)

Thus the time it takes for the expansion fan to reach equi-librium can be more than an order of magnitude longer thanthat required by the contact surface and the shock.

Table 1 Summary of computation cases

Case Y p4 d p (µm) p4/p1 T4/T1

1 0.2 10 5 1

2 0.8 10 5 1

3 0.8 1 20 1

4 0.8 10 20 1

5 0.8 100 20 1

3 Numerical method

The solution method is based on the cell-centered finite-vol-ume method. The following is only a brief overview. Formore details, see [6–10]. The inviscid flux for the gas is cal-culated by the approximate Riemann solver of Roe [16]. Theface-states are obtained by a simplified second-order accu-rate weighted essentially non-oscillatory scheme (see, e.g.,[11]), which modifies the gradients computed using the least-squares reconstruction method of Barth [1].

The method employed to solve the Eulerian equations gov-erning the particle phase variables is very similar to that usedfor the gas. The primary difference is that the inviscid fluxis computed using the method of Larrouturou [13] to ensurepositivity of Y p.

Both the gas and particle variables are evolved in timeusing the fourth-order Runge–Kutta method.

The solution method has been extensively verified and val-idated for gas and gas–particle flows with and without shockwaves. For brevity, the results from these studies are notreproduced here. The studies demonstrated that the expectedorders of convergence and good agreement with experimen-tal data were obtained. Detailed results can be found in thereferences listed above.

4 Results

A wide range of particle mass fractions and particle sizeswere considered to obtain a complete understanding oftheir influence. Here, to illustrate the effect of the particlemass fraction and particle size, we consider the five casesshown in Table 1. Gas and particle properties common to allcases are: C p = 1600 J/(kg K), ρ p = 1800 kg m3, Cg

p =1004.5 J/(kg K), µg = 1.75 × 10−5 kg/(m s), Pr = 0.72,and Rg = 287 J/(kg K).

Additional computations that are not reported here indi-cate that the flow behaviors described in this article are notsignificantly affected by other temperature ratios and dragand heat-transfer laws.

With two exceptions noted below, the mesh resolution andthe time step are 5 × 10−4 m and 8 × 10−7 s, respectively.These values were determined through refinement studies.

123

74 Y. Ling et al.

x (m)

u(m

/s)

-0.0001 -5E-05 0 5E-05 0.0001 0.0001

0

50

100

150

200

ufg

ug

up

(a)Velocity

x (m)

T(K

)

-0.0001 -5E-05 0 5E-05 0.0001 0.0001200

250

300

350

400

Tfg

Tg

Tp

(b) Temperature

Fig. 4 Comparison of computed and pure gas solutions at t = 2 ×10−7 s for case 1, showing that the transient process starts from thepure gas limit

4.1 Overall behavior

Immediately after the breaking of the diaphragm, the puregas solution is established, where the particle velocity andtemperature remain at their initial values and the gas vari-ables are given by the solution of the pure-gas shock-tubeproblem. This behavior is reproduced in the simulations, ascan be seen in Fig. 4 for the velocities and the temperaturesfor case 1. Here, to show the early time behavior, we haveused a mesh resolution and time step of 1 × 10−7 m and1 × 10−10 s, respectively. As t � τ p, the particles equili-brate with the gas and approach the equilibrium solution, as

x (m)

u(m

/s)

-9 -6 -3 0 3 6 9 120

50

100

150

200

ueg

ug

up

(a) Velocity

x (m)

T(k

)

-9 -6 -3 0 3 6 9 12200

250

300

350

400

Teg

Tg

Tp

(b) Temperature

Fig. 5 Comparison of computed and equilibrium solutions at t = 2.5×10−2 s for case 1, showing that the transient process reaches the equi-librium limit

shown in Fig. 5. In the following, we focus on the transientbehavior of the solution, i.e., the way in which the solutionevolves from the pure gas to the equilibrium state.

The overall transient behavior for case 1 is shown inFig. 6 and that for case 2 is shown in Fig. 7. The overalltransient process for case 2 is also a shown in an x − tdiagram, see Fig. 8, which is generated numerically fromthe magnitude of the density gradient. From Fig. 8a, it canbe seen clearly that after long enough time, all the wavesand discontinuities travel at constant speeds, which matchthe equilibrium values. Figure 8b shows the early stages ofthe transient process, from which we can observe how the

123


x (m)

u(m

/s)

-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.0060

50

100

150

200

ug ,t=1e-6sug ,t=5e-6sug ,t=1e-5sup ,t=1e-6sup ,t=5e-6sup ,t=1e-5sug

f ,t=1e-5s

(a)1 × 10− 6 t 1 × 10− 5 s

x (m)

u(m

/s)

-6 -3 0 3 60

50

100

150

200 ug ,t=1e-3sug ,t=5e-3sug ,t=1e-2sup ,t=1e-3sup ,t=5e-3sup ,t=1e-2sug

e ,t=1e-2s

(b)1 × 10− 3 t 1 × 10− 2 s

x (m)

u(m

/s)

-15 -12 -9 -6 -3 0 3 6 9 120

50

100

150

200 ug ,t=1.5e-2sug ,t=2.0e-2sug ,t=2.5e-2sup ,t=1.5e-2sup ,t=2.0e-2sup ,t=2.5e-2sug

e ,t=2.5e-2s

(c)1.5 × 10− 2 t 2.5 × 10− 2 s

Fig. 6 Computed velocity profiles of the transient process for case 1

x (m)

u(m

/s)

-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.0060

50

100

150

200ug ,t=1e-6sug ,t=5e-6sug ,t=1e-5sup ,t=1e-6sup ,t=5e-6sup ,t=1e-5sug

f ,t=1e-5s

(a) 1 × 10− 6 t 1 × 10− 5 s

x (m)

u(m

/s)

-5 -2.5 0 2.5 50

50

100

150ug ,t=1e-3sug ,t=5e-3sug ,t=1e-2sup ,t=1e-3sup ,t=5e-3sup ,t=1e-2sug

e ,t=1e-2s

(b)1 × 10− 3 t 1 × 10− 2 s

x (m)

u(m

/s)

-9 -6 -3 0 3 6 9 120

50

100

150 ug ,t=1.5e-2sug ,t=2.0e-2sug ,t=2.5e-2sup ,t=1.5e-2sup ,t=2.0e-2sup ,t=2.5e-2sug

e ,t=2.5e-2s

(c) 1.5 × 10− 2 t 2.5 × 10− 2 s

Fig. 7 Computed velocity profiles of the transient process for case 2

123

76 Y. Ling et al.

Fig. 8 Numerically generated x − t diagram for case 2, showing thetransient evolution from pure gas limit to equilibrium limit. The com-puted result is shown in solid lines; the pure gas flow result is shown indashed lines. At long times, the computed speeds of the waves matchesthe equilibrium results (boxes). The shading represents the magnitudeof the density gradient

waves match the frozen flow at the very beginning and howthey deviate from it later on.

For ease of exposition, we discuss the evolution of theexpansion fan, particle contact surface, and the shock waveseparately.

4.2 Evolution of expansion wave

Three primary observations can be made about the evolutionof the expansion fan. First, as can be seen from Figs. 6a, 7a,and 8b, and as discussed above, the head of the expansion

fan propagates at constant speed ag4 into the driver section.

Second, and most important given the present focus on theflow transient, the strength of the expansion fan appears tochange rapidly in the very early stages: The velocity andpressure of the gas to the right of the expansion fan decreasevery quickly from the values of the pure gas limit due to thepresence of the particles. This means that the total velocitychange across the expansion fan decreases while the pressurechange increases. In other words, both the internal energy andkinetic energy of the gas phase drop through expansion fan.This is because part of gas energy is transferred to the par-ticles. The total energy of the mixture across the expansionfan is still conserved. Referring to Fig. 7, the change in thestrength of the expansion fan is most clearly visible for largevalues of Y p

4 . It is this change in the expansion fan strengththat sends out disturbances in the form of expansion wavesthat travel in the positive x-direction toward the shock wave.They interact with the particle contact surface, gas contactsurface, and shock wave, causing them to decelerate. In turn,these interactions generate reflected expansion waves thataffect the expansion fan. The third observation concerns theposition of the particle contact surface with respect to thevelocity and temperature profiles in the expansion fan. Aswill be discussed in more detail in the next section, the par-ticle contact surface coincides with the point at which theslope of the velocity profile is discontinuous, as can be seenfrom Figs. 6a and 7a. Thus it is important to note that thepoint at which the velocity profile is discontinuous does notcoincide with the tail of the expansion fan.

The influence of the particle diameter on the initial phaseof the transient behavior of the expansion fan at a high pres-sure ratio can be seen in Fig. 9. For comparison, the puregas solution is also shown. Here, to show the early timebehavior, we have used a mesh resolution and time step of1 × 10−6 m and 1 × 10−9 s, respectively. It is clear that thesmallest particles lead to the largest deviation from the puregas solution. This is easily explained because the drag coef-ficient CD ∝ 1/d p at low particle Reynolds numbers, whileCD ≈ constant at high particle Reynolds numbers. There-fore the total viscous drag force per unit mass is proportionalto 1/(d p)2 at low particle Reynolds numbers, and to 1/d p

at high particle Reynolds numbers. Thus for the same massfraction, larger particles exert a smaller drag force on the gas.As a result, at the very early stages of the transient, the caseswith larger particles are much closer to the pure gas solu-tion. Note also in Fig. 9b the very clearly visible location ofthe particle contact surface and its influence on the velocityprofile in the expansion fan as discussed above.

It can also observed that from Figs. 4, 6c, and 7c that att = 25 ms, both the shock and particle contact surface havereached equilibrium, as the velocity between the shock andtail of expansion fan is constant, implying that the shock andparticle contact surface travel at constant speed. However,

123


x (m)

u(m

/s)

-0.003 -0.0015 0 0.00150

100

200

300

400ug ,t=1e-6sug ,t=3e-6sug ,t=5e-6sug ,t=7e-6sug

f ,t=1e-6sug

f ,t=3e-6sug

f ,t=5e-6sug

f ,t=7e-6s

(a) Case 3

x (m)

u(m

/s)

-0.003 -0.0024 -0.0018 -0.0012 -0.0006 0 0.00060

100

200

300

400

ug ,t=1e-6sug ,t=3e-6sug ,t=5e-6sug ,t=7e-6sug

f ,t=1e-6sug

f ,t=3e-6sug

f ,t=5e-6sug

f ,t=7e-6s

(b) Case 4

x (m)

u(m

/s)

-0.003 -0.0024 -0.0018 -0.0012 -0.0006 0 0.00060

100

200

300

400ug ,t=1e-6sug ,t=3e-6sug ,t=5e-6sug ,t=7e-6sug

f ,t=1e-6sug

f ,t=3e-6sug

f ,t=5e-6sug

f ,t=7e-6s

(c) Case 5

Fig. 9 Computed velocity profiles in the expansion fan at the earlystages of the transient process for 1 × 10−6 � t � 7 × 10−6 s

near the head and tail of expansion fan, the waves are stilltraveling with speeds different from their equilibrium values,which means that the expansion fan has not fully reachedequilibrium. Therefore, the equilibration time for expansionfan is much longer than for the shock, which is consistentwith the time scale analysis in the previous section.

4.3 Evolution of particle contact surface

Because of the above-described change of strength of theexpansion fan, it is clear that any disturbances introducedby this change will travel at the speed u + a in the formof expansion waves toward the particle contact surface. Theinteraction with the expansion waves causes the particle con-tact surface to decelerate.

Effect of particle mass fraction The temporal evolution ofthe gas and particle velocities at the particle-contact surfacelocation are plotted in Fig. 10 as a function of Y p

4 for variousd p. A critical value of Y p

4 appears to exist, above which theapproach of the particle velocity at the particle contact surfaceto the equilibrium velocity is non-monotonic; below that crit-ical value, the approach of the gas velocity is non-monotonic.The non-monotonic behavior of particle velocity will be dis-cussed below, and similar behavior is seen for the speed ofthe shock wave. The variation of the critical value of Y p

4 withd p determined numerically, is presented in Fig. 11. When theparticle mass fraction is close to the critical value, the ampli-tudes of the over- or undershoots are small or barely visible.On the other hand, when the particle mass fraction is far fromthe critical value, such as Y p

4 = 0.1 and Y p4 = 0.8 in Fig. 10b,

the non-monotonic behavior can be observed clearly.A possible explanation for the non-monotonic behavior

is as follows. For a specific particle size, if the mass frac-tion is very low, the equilibrium value of the contact surfacevelocity is very close to the pure gas value, therefore the timefor the gas velocity at the contact surface to decrease to itsequilibrium value is much shorter than that for the particlevelocity to reach its equilibrium value. So when the gas veloc-ity decreases to the equilibrium value, the particle velocityis still smaller than the equilibrium value, and therefore theviscous drag force will cause the gas velocity to continue todecrease and undershoot its equilibrium value. On the otherhand, when the mass fraction is very high, the differencebetween equilibrium value and pure gas value of the gasvelocity at the contact surface is much larger, and the timefor the gas velocity to reach equilibrium value increases. Incontrast to the low mass fraction case, the particle velocity atthe contact surface reaches the equilibrium value before thegas, so the viscous drag force increases the particle velocityto overshoot the equilibrium value. It is interesting to find thatexcept in the vicinity of the critical value, either the gas or the

123

78 Y. Ling et al.

t (s)

ug ,up

(m/s

)

0 2E-05 4E-05 6E-05120

140

160

180

2000.00.10.20.3

0.4

0.5

0.6

0.7

0.8

Yp4

(a) d p = 1 µ m

t (s)

ug ,up

(m/s

)

0 0.002 0.004 0.006120

140

160

180

2000.00.10.20.3

0.4

0.5

0.6

0.7

0.8

Yp4

(b) d p = 1 0 µ m

t (s)

ug ,up

(m/s

)

0 0.02 0.04 0.06120

140

160

180

2000.00.10.2

0.3

0.4

0.5

0.6

0.7

0.8

Yp4

(c) d p = 100 µ m

Fig. 10 Comparison of computed evolution of gas (dash-dotted lines)and particle (solid lines) velocities at particle contact surface locationas a function of Y p

4 for various values of d p

log(dp)

Yp 4

-6 -5.5 -5 -4.5 -4 -3.50.2

0.3

0.4

0.5

0.6

0.7

Yp4,c

Yp4,s

us monotonic

us non-monotonicup monotonic

up non-monotonic

Fig. 11 Variation of Y p4 with d p for which approach to equilibrium of

gas and particle velocities at particle contact surface and shock velocityis non-monotonic

particle velocity have to experience non-monotonic behaviorto approach equilibrium. We have verified that these behav-iors are not caused by the specific drag and heat-transfercorrelation or the initial conditions used in the simulations.

Effect of particle size The temporal evolution of the gas andparticle velocities and temperatures at the particle contactsurface location are plotted in Figs. 12 and 13 as a functionof d p for Y p

4 = 0.3 and 0.8, respectively. The followingpoints can be made. First, it is observed that if the time isscaled by particle mechanical response time τ p, both themechanical and thermal equilibration times for different sizeparticles are of the same order, implying that the effect ofthe particle size is reflected mainly through τ p. Second, itcan be found from Fig. 12a that as the particle size increases,the evolution of the gas velocity at the particle contact sur-face location changes from monotonic to non-monotonic forY p

4 = 0.3. From Fig. 13a, we observe that the non-mono-tonic behavior occurs for all three particle sizes, in which theparticle velocities rise above the gas velocities and the equi-librium values and then drop back to the latter. Third, in bothcases, the dimensionless mechanical and thermal equilibra-tion times t∗c /τ p decrease as particle size increases. Finally,for Y p

4 = 0.3, the thermal equilibration time scale is approx-imately equal to the mechanical equilibration time scale forvarious particle sizes; but for Y p

4 = 0.8, the thermal equili-bration time scale is obviously smaller than the mechanicalequilibration time scale for various particle sizes. It shouldalso be noted that the computed equilibrium temperature val-ues are slightly different for the various values of d p in bothFigs. 12b and 13b. The reason is that the particle contact

123


t/τp

ug ,up

(m/s

)

0 1 2 3 4 5 6175

180

185

190

195

up,dp=3µmug,dp=3µmup,dp=10µmug,dp=10µmup,dp=30µmug,dp=30µm

(a) Velocity

t/τp

Tg ,T

p(K

)

0 1 2 3 4 5 6240

250

260

270

280

Tp,dp=3µmTg,dp=3µmTp,dp=10µmTg,dp=10µmTp,dp=30µmTg,dp=30µm

(b) Temperature

Fig. 12 Comparison of computed evolution of gas (dash-dotted lines)and particle (solid lines) velocities and temperatures at particle contactsurface location as a function of d p for Y p

4 = 0.3

surface is very close to the gas contact surface, where thenumerical temperature gradient is very large. Therefore, theslight differences in Figs. 12b and 13b are due to extractingthe values close to a near-discontinuity.

Comparison with time-scale analysis From Fig. 10, the timescale to reach equilibrium conditions can be observed. Theresults for the computed equilibration time scales are plottedin Fig. 14 as a function of the particle diameter. For com-parison, we can also estimate the equilibration time scale ofthe particle contact surface from (35) for cases with differentmass fractions and particle sizes. The results for Y p

4 = 0.2

t/τp

ug ,up

(m/s

)

0 2 4 6130

135

140

145

150

up,dp=3µmug,dp=3µmup,dp=10µmug,dp=10µmup,dp=30µmug,dp=30µm

(a) Velocity

t/τp

Tg ,T

p(K

)

0 1 2 3 4 5 6265

270

275

280

285

Tp,dp=3µmTg,dp=3µmTp,dp=10µmTg,dp=10µmTp,dp=30µmTg,dp=30µm

(b) Temperature

Fig. 13 Comparison of computed evolution of gas (dash-dotted lines)and particle (solid lines) velocities and temperatures at particle contactsurface location as a function of d p for Y p

4 = 0.8

and 0.8, d p = 1 and 100 µm are shown in Table 2. From theseresults, we can estimate the value of d[log(t∗c )]/d[log(d p)],and compare with the computed results presented in Fig. 14.For Y p

4 = 0.2, over the range d p = 1–100 µm, we obtaind[log(t∗c )]/d[log(d p)] = 1.645; and for Y p

4 = 0.8, we obtaind[log(t∗c )]/d[log(d p)] = 1.671. The dependence on Y p

4 isweak, and the time scale analysis predicts reasonably wellthe computed power-law behavior of t∗c ∝ (d p)1.66.

It should also be mentioned here that the particle contactsurface always travels close to the gas contact surface becausethe particles studied here are very small. After equilibrium isreached, the particles and gas contact surface travel with the

123

80 Y. Ling et al.

log(dp)

log(

t* s),l

og(t

* c)

-6 -5.5 -5 -4.5 -4-5

-4

-3

-2

-1

0 t*s ,Yp=0.2

t*s ,Yp=0.8

t*c ,Yp=0.2

t*c ,Yp=0.8

2.05

1

1

1.66

Fig. 14 Equilibrium time scales of shock and particle contact surfaceas a function of d p for Y p

4 = 0.2 and 0.8 from computations, comparedwith the result from the time scale analysis indicated in slope symbol

Table 2 Particle contact surface equilibration time scales as given bytime-scale analysis

Y p4 d p (µm) p4/p1 T4/T1 t p∗ log(t p∗)

0.2 1 5 1 1.846E−5 −4.734

0.2 100 5 1 3.602E−2 −1.443

0.8 1 5 1 1.955E−5 −4.709

0.8 100 5 1 4.304E−2 −1.366

same speed (see Fig. 8b) and therefore the distance betweenthe particle and gas contact surfaces becomes constant.

4.4 Evolution of shock wave

Similar to the evolution of the particle contact surface, theexpansion waves sent by the change of strength of the expan-sion fan causes the shock wave to decelerate as alreadydiscussed above. The evolution of the shock speed is pre-sented in Fig. 15 as a function of d p and Y p

4 . As with thecorresponding evolution of the velocities at the particle con-tact surface, we note that the asymptotic approach of theshock speed to its equilibrium value is not monotonic if themass fraction Y p

4 is below a critical value for a given particlediameter d p. The critical values for the shock wave are quitesimilar to those for the particle contact surface over a rangeof d p, see Fig. 11.

It is interesting to note that, when the shock speed is non-monotonic, the shock is at first moving faster, then slowerthan the equilibrium value. It is therefore possible to predictthe shock location to be very close to the equilibrium valuefor certain Y p

4 and d p. This means that even for finite-sized

t (s)

u s(m

/s)

0 2E-05 4E-05 6E-05 8E-05 0.0001430

440

450

460

470

4800.00.10.2

0.4

0.3

0.5

0.6

0.7

0.8

Yp4

(a) d p = 1 µ m

t (s)

u s(m

/s)

0 0.002 0.004 0.006 0.008 0.01430

440

450

460

470

4800.0Yp

4

0.8

0.7

0.6

0.5

0.4

0.3

0.20.1

(b) d p = 10 µ m

t (s)

u s(m

/s)

0 0.02 0.04 0.06 0.08 0.1430

440

450

460

470

4800.00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Yp4

(c) d p = 100 µ m

Fig. 15 Comparison of computed evolution of shock speed as a func-tion of Y p

4 , (shock speed of equilibrium flow solution is showed asdashed lines)

123


Table 3 Shock equilibration time scales as given by time-scale analysis

Y p4 d p (µm) p4/p1 T4/T1 t∗s log(t∗s )

0.2 1 5 1 2.185E−5 −4.661

0.2 100 5 1 2.847E−1 −0.546

0.8 1 5 1 1.685E−5 −4.773

0.8 100 5 1 2.134E−1 −0.671

particle, equilibrium theory can predict the shock locationaccurately after equilibrium conditions are reached.

Comparison of Figs. 10 and 15 shows that the time for theshock wave to reach equilibrium is larger than that for the par-ticle contact surface. This can also be observed from Fig. 14,which shows the equilibration time scales of the shock waveand particle contact surface for d p varying from 1 − 100 µmfor Y p

4 = 0.2 and 0.8.For comparison, we can also estimate the equilibration

time scale of the shock from (40) for different mass frac-tions and particle sizes. The results for Y p

4 = 0.2 and 0.8and d p = 1 and 100 µm are shown in Table 3. From theseresults, we can estimate the value of d[log(t∗s )]/d[log(d p)],and compare with the computed results (see Fig. 14). ForY p

4 = 0.2, over the range d p = 1–100 µm, we obtaind[log(t∗s )]/d[log(d p)] = 2.057; and for Y p

4 = 0.8, we obtaind[log(t∗s )]/d[log(d p)] = 2.051. Similar to the results for theparticle contact surface, the dependence on Y p

4 is weak andthe time scale analysis predicts quite accurately the computedpower-law behavior of t∗s ∝ (d p)2.05.

5 Conclusions

The present study focused on the shock-tube problem to studythe gas–particle flow arising from particle-laden driver sec-tions. Theory and computation were used to understand theinteraction between the expansion fan and particles. The tran-sient process from frozen to equilibrium flow was investi-gated in detail. The mechanism by which particles changethe gas flow was analyzed by studying the propagations ofdisturbance waves.

Simple theoretical considerations allow estimation of thetime scales with which the expansion fan, the particle con-tact, and the shock wave reach equilibrium. These estimatesindicated that the longest equilibration time scale is associ-ated with the expansion fan. Numerical results were in goodagreement with the theoretical estimates.

The particle mass fraction was found to determine whetherthe approach to equilibrium conditions of the shock and par-ticle contact-surface speeds are monotonic or not. We havefound that for values of the mass fraction below a diameter-

dependent critical value, the approach to equilibrium condi-tions is not monotonic.

The initial temperature ratio and the drag and heat-transferlaws do not affect our results significantly.

Acknowledgments The authors gratefully acknowledge support bythe National Science Foundation under grant number EAR0609712.

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Transient phenomena in one-dimensional compressible gas–particle flows

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