Download - Calculation of subsonic gas-dispersion flows in curved channels by the particle-in-cell method

ISSN 0040-5795, Theoretical Foundations of Chemical Engineering, 2006, Vol. 40, No. 1, pp. 80–88. © Pleiades Publishing, Inc., 2006.Original Russian Text © I.Kh. Enikeev, 2006, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2006, Vol. 40, No. 1, pp. 85–94.

80

Let us consider a flow of a two-phase medium con-sisting of a compressible carrier gas and monodispersedrops in a curved channel of width

R

and finite length

L

.Figure 1 presents a half of the longitudinal section ofthe channel. Let the symmetry axis be the

ox

axis of acylindrical coordinate system (

x

,

y

,

ϕ

). Then, one canwrite the equations of the unsteady-state two-dimen-sional flow of the two-phase medium in the channel interms of a model of interpenetrating continua in dimen-sionless form [1, 2]:

(1)

Set (1) should be complemented with equations ofstate of the phases; it is assumed that the carriermedium is a perfect gas and the drops are incompress-

ible ( = const):

∂ρs

∂t-------- ∇k ρsvs

k( )+ 0,=

∂ρsvs

∂t------------- ∇k ρsvsvs

k( )+δα---∇ p– F,+=

α γ M02, δ

1, s 1=

0, s 2,=⎩⎨⎧

= =

∂ρsEs

∂t-------------- ∇k ρsEs δρ/α+( )v s

k[ ]+⎩ ⎭⎨ ⎬⎧ ⎫

s 1=

2

∑ 0,=

Es es12--- vs vs,( ),+=

∂ρses

∂t------------- ∇k ρsesvs

k( )+ q 1/2F v1 v2–( ).+=

ρ20

(2)

The dimensionless parameters in Eqs. (1) and (2) canbe represented as

(3)

The vector

F

of the friction force between the phasesand the interphase heat-transfer intensity

q

areexpressed as [1, 2]

(4)

To integrate Eqs. (1) and (2), it is necessary toimpose boundary and initial conditions. Let us assumethat the left boundary of the region from which the gas–drop medium flows is far enough. Then, as

x

–

∞

,there is a flow without dynamic (velocity) lag or ther-mal (temperature) lag of drops in which the verticalcomponents of the gas and drop velocities are zero. Inthis case, the flow uniformity condition [3–5]

p α γ 1–( )ρ1e1,=

e1

T1

α γ 1–( )--------------------, e2–

c2T2

αc γ 1–( )-----------------------.= =

vs

vs( )*U

------------, ρρs*

p10-------, ρ ρ*

ρ0------,= = =

Es

Es*

U02

-------, es

es*

U02

------, tU0

R------t*,= = =

xx*R------, y

y*R------.= =

F Stρ2 v1 v2–( )ρ2, Stρ2

0U0d2

18µR------------------,= =

q β e1cc2----e2–⎝ ⎠

⎛ ⎞ ρ2, β 12λR

ρ20U0d2c

---------------------.= =

∂v 1x

∂x--------- 0,=

Calculation of Subsonic Gas-Dispersion Flowsin Curved Channels by the Particle-in-Cell Method

I. Kh. EnikeevMoscow State University of Engineering Ecology,

Staraya Basmannaya ul. 21/4, Moscow, 107884 RussiaReceived March 1, 2005

Abstract—Multiphase (specifically, two-phase) flows of media in complex-shaped channels are studied by theparticle-in-cell method. The stability of the proposed algorithm is analyzed over a wide range of Mach numbersof an undisturbed flow. The use of the method is exemplified by calculating the characteristics of a gas–dropflow in curved channels in which the generatrices of the lateral surface consist of circular arcs.

DOI: 10.1134/S0040579506010118

THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 40 No. 1 2006

CALCULATION OF SUBSONIC GAS-DISPERSION FLOWS 81

where is the projection of the gas velocity on theaxis x, can be supposed to be met.

It is also assumed that the flow of the mixture in thisregion is isentropic and isenthalpic, i.e., H0 = const andS = const. The reduced density of the second phase inthis region of the channel is a given value. In the calcu-lations, these boundary conditions are imposed in thesection x = –1.25 (Fig. 1).

It is supposed that the gas does not penetrate the sidewalls and that drops that have arrived at the channelwalls adhere to them and thus leave the flow. Taken asthe boundary conditions for the gas at the outlet of thechannel are the relations obtained for a one-dimen-sional isentropic gas flow from a nozzle [6]

v 1x

v1x

v1 1( )----------

MM1-------

1 0.5 γ 1–( )M 1( )2+

1 0.5 γ 1–( )M2+-------------------------------------------- ,=

As the initial data, the parameters of the undisturbedflow in the section x = –1.25 at F = 0 and q = 0 are taken.The set of Eqs. (1) is integrated by the finite-differencemethod in the curvilinear region M:

where G(x) and F(x) are the equations of the lower andupper lateral surfaces of the channel. In the integration,it is necessary to replace the continuous curvilinearregion M by a grid. The direct replacement gives rise toirregular nodes or computational cells near the bound-ary of the region. In this case, to impose the boundaryconditions, fractional cells are formed in the layer ofirregular computational cells using the algorithm of theparticle-in-cell method [7]. The practice of calculationsusing fractional cells showed that this algorithm is quitecumbersome, especially in the case where M0 � 1.Therefore, it is more expedient to introduce such newvariables ξ = ξ(x, y) and η = η(x, y) that the curvilinearregion becomes rectangular. As was shown [3, 4], if theJacobian I = D(ξ, η)/D(x, y) of this transformationexists and is nonzero in any point of the region, then thedivergent form of Eqs. (1) is retained. By changing theindependent variables x = x,

the curvilinear region M is transformed into the rectan-gular region N:

Then, in the variables (x, ξ), the set of Eqs. (1) takes theform

(5)

AA 1( )---------

MM1-------

1 0.5 γ 1–( )M 1( )2+

1 0.5 γ 1–( )M2+-------------------------------------------- .=

M 0 x L/R G x( ) y F x( )≤ ≤,≤ ≤{ },

ξ y G x( )–F x( ) G x( )–------------------------------,=

N 0 x 1 0 ξ 1≤ ≤,≤ ≤( ).

∂ρs

∂t--------

∂ρsv sx

∂x--------------

1ε---

∂ρsUs

ε∂ξ--------------+ +

ρsUsε

ε εξ G+( )------------------------,–=

∂ρsv sx

∂t--------------

∂ρs v sx( )2

∂x---------------------

1ε---

∂ρsv sUs

∂ξ---------------------+ +

= 1

αε------ ε∂p

∂x------– εξ G'+( )∂ρ

∂ξ------+

ρsv sxUs

ε

ε εξ G+( )------------------------ 1–( )sFx,+–

∂ρsv sy

∂t--------------

∂ρsv syv s

x

∂x---------------------

1ε---

∂ρsv syUs

∂ξ---------------------+ +

= 1

αξ-------∂p

∂ξ------– 1–( )sFy ρsv s

yUsε

ε εξ G+( )------------------------,–+

(c)

–1.0 –0.5 0 0.5x/R

y/R

0.3

0.6

0.9 1.1

(b)

(a)

0.2

0.4 0.7 0.8

abcd

0.040.07

0.11 0.12

Fig. 1. Level curves (a) calculated in this work and (b)determined in the literature [5] and the (c) gas and (d) dropstreamlines in a convergent tube at M0 = (a) 0.08, (b) 0.5,and (c) 0.9.

82


ENIKEEV

If = ε' (εξ + G), then set (5) appears as a set ofequations for the planar case.

As was shown [7, 8] for flows in rectangular regionsat å0 � 1, at the Euler step, it is better to use a time-implicit difference scheme for calculating the pressure.

In this work, this method is generalized to regions ofcomplex shape.

Let us write the equations describing the gas flow atthe Euler step in the form

(6)

In derivation of Eqs. (6), the terms taking intoaccount the axial symmetry and convective transfer andalso interphase interaction are ignored. The region ofintegration is covered with a fixed computational gridwith rectangular cells with sides ∆x and ∆ξ. Let us labelthe centers of the cells with subscripts i and j along thecoordinates x and ξ, respectively, and write Eq. (6) indifference form according to the implicit scheme

(7)

where χ = dt/dx,

(8)

∂ρsEs

∂t--------------

∂ ρsEs δ/αp+( )v sx

∂x--------------------------------------------+

s 1=

2

∑

+1ε---

∂ ρsUs δ/αp+( )Us

∂ξ---------------------------------------------

δp/α ρsEs+( )Us

ε εξ G+( )----------------------------------------+ 0,=

∂ρse2

∂t-------------

∂ρ2e2v 2x

∂x--------------------

1ε---

∂ρ2e2U2

ε∂ξ--------------------+ +

= qρ2e2U2

ε

ε εξ G+( )------------------------–

12---F v1 v2–( ),+

Usε εv s

y ε'v sx εξ G+( ),+=

Us v sy ε'v G'+( )v s

x, ε x( )– F x( ) G x( ).–= =

Usε

v sx

∂u∂t------

1αρ-------∂p

∂x------–

ε'ξ G'+( )αερ

-----------------------∂p∂ξ------,+=

∂v∂t-------

1αερ----------∂p

∂ξ------,–=

∂p∂t------

γ 1–( )ε

---------------- p∂εu∂x

--------- ∂ ε'ξ G'+( )u∂ξ

------------------------------+ .–=

ui j,n 1+ ui j,

n χαρi j,

n------------ pi 1/2 j,+

n 1+ pi 1/2 j,–n 1+–( )–=

+χ εi'ξ j Gi'+( )

αεiρi j,n

----------------------------- pi j 1/2+,n 1+ pi j 1/2–,

n 1+–( ),

v i j,n 1+

v i j,n χ

αεiρi j,n

---------------- pi j 1/2+,n 1+ pi j 1/2–,

n 1+–( ),–=

(9)

In Eqs. (7)–(9), the parameters of the gas at theboundaries of cells are approximated by central differ-ences by formulas of the first order of accuracy, e.g.,

Let us eliminate and at the (n + 1)th timestep from Eq. (9) using Eqs. (7) and (8) and introducethe notation

pi j,n 1+ pi j,

n γ 1–( )χ pi j,n 1+

εi

-------------------------------- εi ui 1/2 j,+n 1+ ui 1/2– j,

n 1+–( )[–=

+ v i j 1/2+,n 1+

v i j 1/2–,n 1+–

– εi' ξ j 1/2+ ui j 1/2+,n 1+ ξ j 1/2– ui j 1/2–,

n 1+–( )

– G' ui j 1/2+,n 1+ ui j 1/2–,

n 1+–( ) ].

ui 1/2 j,+n 1+ 1

2--- ui 1 j,+

n 1+ ui j,n 1++( ).=

ui j,n 1+

v i j,n 1+

Aiχ2 γ 1–( )

αεi

----------------------ρi j,n 1+=

×εi 1/2–

ρi 1/2 j,–n

-----------------14---

Gi' ξ j 1/2+ εi'+

ρi j 1/2+,n

------------------------------Gi' ξ j 1/2– εi'+

ρi j 1/2–,n

------------------------------–⎝ ⎠⎛ ⎞+ ,

Biχ2 γ 1–( )

αεi

----------------------ρi j,n 1+=

×εi 1/2+

ρi 1/2 j,+n

-----------------14---

Gi' ξ j 1/2+ εi'+

ρi j 1/2+,n

------------------------------Gi' ξ j 1/2– εi'+

ρi j 1/2–,n

------------------------------–⎝ ⎠⎛ ⎞– ,

Ci 1 Ai Biχ2 γ 1–( )

αεi2

----------------------ρi j,n 1++ + +

⎩⎨⎧

=

×1 Gi' ξ j 1/2+ εi'+( )2

+

ρi j 1/2+,n

----------------------------------------------1 Gi' ξ j 1/2– εi'+( )2

+

ρi j 1/2–,n

----------------------------------------------+⎭⎬⎫

,

A j*χ2 γ 1–( ) pi j,

n 1+

αεi

----------------------------------1 Gi' ξ j 1/2– εi'+( )2

+

εiρi j, 1/2–n

----------------------------------------------=

+14---

Gi 1/2+' ξ jεi 1/2+'+

ρi 1/2 j,+n

---------------------------------------Gi 1/2–' ξ jεi 1/2–'+

ρi 1/2 j,–n

---------------------------------------–⎝ ⎠⎛ ⎞ ,

B j*χ2 γ 1–( ) pi j,

n 1+

αεi

----------------------------------1 Gi' ξ j 1/2+ εi'+( )2

+

εiρi j, 1/2+n

----------------------------------------------=

–14---

Gi 1/2+' ξ jεi 1/2+'+

ρi 1/2 j,+n

---------------------------------------Gi 1/2–' ξ jεi 1/2–'+

ρi 1/2 j,–n

---------------------------------------–⎝ ⎠⎛ ⎞ ,



In this notation, Eq. (9) is written as follows:for the longitudinal sweep,

(10)

for the transverse sweep,

(11)

Since the sweep coefficients in Eqs. (10) and (11)

depend on , these equations are solved by iteration

in .

At each iteration, the values in the expressions

for Ai, Bi, , , Ci, Cj, Fi, and are replaced bythe respective values obtained at the preceding itera-tion. After solving set (7)–(9) with a given accuracy, atthe Euler step, the pressure and velocities of the carrierphase are found. All the parameters of the dispersedphase at the Euler step remain unchanged with time.

C j* 1 A j* B j*+ +=

+χ2 γ 1–( )

αεi

----------------------ρi j,n 1+ εi 1/2+'

ρi 1/2 j,+n

-----------------εi 1/2–'

ρi 1/2– j,n

-----------------+⎝ ⎠⎛ ⎞ ,

Φi*χ2 γ 1–( ) pi j,

n 1+

αεi

----------------------------------=

×Gi 1/2+' ξ jεi 1/2+'+

ρi 1/2 j,+n

---------------------------------------Gi' ξ j 1/2+ εi'+

ρi j 1/2+,n

------------------------------+⎝ ⎠⎛ ⎞ ρi 1+ j 1+,

n 1+

–Gi 1/2+' ξ jεi 1/2+'+

ρi 1/2 j,+n

---------------------------------------Gi' ξ j 1/2– εi'+

ρi j 1/2–,n

------------------------------+⎝ ⎠⎛ ⎞ ρi 1 j 1–,+

n 1+

–Gi 1/2–' ξ jεi 1/2–'+

ρi 1/2 j,–n

---------------------------------------Gi' ξ j 1/2+ εi'+

ρi j 1/2+,------------------------------+⎝ ⎠

⎛ ⎞ ρi 1 j 1+,–n 1+

+Gi 1/2–' ξ jεi 1/2–'+

ρi 1/2– j,n

---------------------------------------Gi' ξ j 1/2– εi'+

ρi j 1/2–,------------------------------+⎝ ⎠

⎛ ⎞ ρi 1 j 1+,–n 1+

+χεi

--- εi 1/2+ ui 1/2 j,+n εi 1/2– ui 1/2 j,–

n– v i j 1/2+,n+[

– v i j 1/2–,n G' ξ j 1/2+ εii'+( )ui j 1/2+,

n–

+ G' ξ j 1/2– εii'+( )ui j 1/2–,n ],

Fi pi j,n– A j* pi j 1–,

n 1+– B j* pi j 1+,n 1+– Φi,+=

F j* pi j,n A j pi 1 j,–

n 1+– B j* pi 1 j,+n 1+– Φ j*,+–=

Φi Φ j*.=

Ai pi 1 j,–n 1+ Ci pi j,

n 1+– Bi pi 1 j,+n 1++ Φi;=

A j* pi j 1–,n 1+ C j* pi j,

n 1+– B j* pi j 1+,n 1++ Φ j*.=

pi j,n 1+

pi j,n 1+

pi j,n 1+

A j* B j* F j*

The Lagrange and final steps are performed by theparticle-in-cell method [7].

Let us consider the stability of the proposed differ-ence scheme using the results [7, 8] of analyzing theparabolic form of the first differential approximation ofthe scheme.

Since inherent pressure in the medium of particles isabsent, the equations of motion of the dispersed phaseare integrated according to an explicit scheme of theparticle-in-cell method [7]. The scheme is well knownin the literature [7, 8]; therefore, the correspondingequations in the initial set of Eqs. (1) are not consideredhere.

Let us consider the case where, for several charac-teristic times of the problem, the following relations arevalid:

Then, the terms describing the intensity of the forceand thermal interphase interaction generally do notaffect the stability of the numerical scheme since theydo not contain higher derivatives. Therefore, the formu-las of the final step for the gas phase ignore the inter-phase interaction forces.

When max(τF, τq) � τ0, an equilibrium scheme ofthe flow of the two-phase medium takes place, in whichthe gas-dispersion flow is regarded as an effective gaswith a changed sound velocity and adiabatic exponent.Therefore, in this case, the interphase interaction forcescan also be ignored. Without loss of generality, the flowis assumed to be plane. Thus, under the assumptionsmade, the equations of the final stage for the gas havethe form

τ0 max τF τq,( ),≅

τ0 R/U0, τF lF/U0, τq lq/U0.= = =

ρi j,n 1+ ρi j,

n ∆t2∆x----------+=

× ρi 1 j,–n ui 1 j,– ui j,+( ) ρi j,

n ui j, ui j,+( ) -–⎩⎨⎧

+1ε--- ρi j 1–,

n Ui j 1–, Ui j,+( ) ρi j,n Ui j, Ui j 1+,+( )–[ ]

⎭⎬⎫

– ρi j,n Ui j,

ε ∆t,

ρi j,n 1+ ui j,

n 1+ = ρi j,n ui j,

∆t2∆x---------- ρi 1 j,–

n ui 1 j,– ui 1 j,– ui j,+( ) ---⎩⎨⎧

+

– ρi j,n ui j, ui j, ui 1 j,++( )

1εi

--- ρi j 1–,n ui j 1–, Ui j 1–, Ui j,+( )[+

84


ENIKEEV

(12)

Here, it is assumed that + > 0, +

> 0, and so on. Obviously, the choice of the flowdirection does not affect the character of the stability ofthe scheme. Let us substitute expressions (7)–(9) for thevelocity components and and the pressure p intoformulas (12) of the final step.

Supposing that all the functions in the obtained rela-tions are continuously differentiable functions of thevariables t, x, and y, let us expand them in Taylor seriesin the vicinity of the point (tn + 1, xi, yj) and truncatethem after the terms of the second order in ∆t, ∆x, and∆y. Thus, the parabolic form of the first differentialapproximation of the initial equations is obtained:

---– ρi j,n ui j, Ui j, Ui j 1+,+( ) ]

⎭⎬⎫

ρi j,n ui j, Ui j,

ε ∆t,–

ρi j,n 1+

v i j,n 1+ = ρi j,

nv i j,

∆t2∆x---------- ρi 1 j,–

nv i 1 j,– ui 1 j,– ui j,+( ) -

⎩⎨⎧

+

– ρi j,nv i j, ui j, ui 1 j,++( )

1εi

--- ρi j 1–,n

v i j 1–, Ui j 1–, Ui j,+( )[+

--– ρi j,nv i j, Ui j, Ui j 1+,+( ) ]

⎭⎬⎫

ρi j,nv i j, Ui j,

ε ∆t,–

ρi j,n 1+ Ei j,

n 1+ = ρi j,n Ei j,

∆t2∆x---------- ρi 1 j,–

n Ei 1 j,– ui 1 j,– ui j,+( ) -⎩⎨⎧

+

– ρi j,n Ei j, ui j, ui 1 j,++( )

1εi

--- ρi j 1–,n Ei j 1–, Ui j 1–, Ui j,+( )[+

--– ρi j,n Ei j, Ui j, Ui j 1+,+( ) ]

⎭⎬⎫

ρi j, Ei j, Ui j,ε ∆t,–

Ui j,ε

ui j, εi' εiξ j Gi+( ) εiv i j, .+=

ui 1 j,– ui j, Ui j 1–,

Ui j,

u v

∂ρ∂t------ ∂ρu

∂x---------

1ε---∂ρU

∂ξ-----------+ +

= 12--- u∆x

1 ε2 ε'2ξ2+ +( )αρε2

------------------------------------ p u2– ∆t+⎩ ⎭⎨ ⎬⎧ ⎫∂2ρ

∂x2--------

– uUε'ξραρ

----------+⎝ ⎠⎛ ⎞ ∆t

ε-----∂2ρU

∂x∂ξ-------------

+12--- U∆x

∆tε

----- ε'ξ∆xαρ

--------------∂p∂x------ 1 ε'2ξ2+( )

αρ------------------------- p

U2

ε------–++

⎩ ⎭⎨ ⎬⎧ ⎫

(13)

× ∂2 p

∂2ξ-------- ∆p,+

∂ρu∂t

--------- ∂ρu2

∂x------------ ∂ρuU

ε∂ξ--------------

1α--- ∂p

∂x------

ε'ξ G'+( )ε

-----------------------∂p∂ξ------–+ + +

= 12--- ρU∆x

γpα----- 2γ 2 5γ– 6+( )ρu2– ∆t+

⎩ ⎭⎨ ⎬⎧ ⎫∂2u

∂x2--------

+ ε'ξγpα

-------------ρu2

------ uε'ξ 5 2γ–( ) U+[+–⎩⎨⎧

+ γ 1–( ) U u–( ) γ 1–( ) 2γ 1–( ) U ε'ξu–( )+⎭⎬⎫

× ∆tε

----- ∂2u∂x∂ξ------------ 1

2ε----- ρU∆x

γpε'2ξ2

α-----------------+

⎩⎨⎧

+

+ ρu ε'ξ U 2uε'ξ–( )[

--– γ 1–( ) u 1 ε'2ξ2+( ) Uε'ξ–( ) ] ∆tε

-----⎭⎬⎫∂2u

∂ξ2-------- ∆u,+

∂ρν∂t

---------- ∂ρuv∂x

--------------1ε--- ∂

∂ξ------ ρuU

pα---+⎝ ⎠

⎛ ⎞+ +

= ρ2--- u∆x u2 γ 1–( )v 2+[ ]∆t–{ }∂2

v

∂x2---------

– ρv 2U 4 γ–( )u 3ε'ξ γ 1–( )v+ +{

+ γ 1–( ) 2γ 1–( )u }∆tε

----- ∂2v

∂x∂ξ------------

+12--- ρU∆x

1

ε2---- ρv 0.5 U 2v+( )[ -–+

⎩⎨⎧

+ γ 1–( ) 2γ 1–( )U ] γpα-----+ ∆t

⎭⎬⎫∂2

v

∂ξ2--------- ∆v ,+

∂ρE∂t

---------- ∂ ρE p+( )u∂x

----------------------------∂ ρE p+( )U

ε∂ξ------------------------------+ +

= ρ u∆x2

---------- γ 1–( )E γ 2 γ 1–+( )u2+[ ]∆t+⎩ ⎭⎨ ⎬⎧ ⎫∂2E

∂x2---------



As was shown [7, 8], to analyze the stability of thescheme, it is necessary to consider the expressions forthe coefficients in front of the second derivatives ∂2/∂x2

and ∂2/∂ξ2 in each of the equations (e.g., for the para-bolic form of the differential approximation of the con-tinuity equation, the stability criteria follow from therelations for the coefficients in front of the derivatives∂2ρ/∂x2 and ∂2ρ/∂ξ2). The coefficients are the diagonalelements of the approximate viscosity matrix [8]. Forbrevity, let us designate the left-hand sides of the initialset of Eqs. (13) by the symbol L( f ), where f is one ofthe sought variables written in the following order:

(14)

The subscripts of the parameters of the gas phase aredropped. As already noted, the difference scheme forthe equations for the dispersed phase is explicit. There-fore, the diagonal elements of the approximate viscos-ity coefficient matrix for this phase are not investigatedsince they were considered in detail previously [8].

The parabolic form of the first differential approxi-mation of the difference scheme can be represented as

where Ä, Ç, and ë are the approximate viscosity coef-ficient matrices and ∆f are the terms containing no sec-ond derivatives of the variable f. The expressions for thediagonal elements of the matrices A, B, and C have theform

+2ρUuγ 2∆t

ε-------------------------- ∂2E

∂ε∂ξ------------

+ρε--- U∆x

2-----------

∆tε

----- γ 2 γ 1–+( )U2 1 ε'2ξ2+( )E+[ ]+⎩ ⎭⎨ ⎬⎧ ⎫

× ∂2E

∂ξ2--------- ∆E.+

ρ u v E., , ,

L f( ) A∂2 f

∂x2-------- B

∂2 f

∂ξ2-------- C

∂2 f∂x∂ξ------------ ∆ f ,+ + +=

A11ρ 1

2--- u∆x

p 1 ε2 ε'2ξ2+ +( )αρε2

---------------------------------------- u2– ∆t+⎩ ⎭⎨ ⎬⎧ ⎫

,=

B11ρ 1

2--- U∆x

ε'ξ∆xαρ

--------------∂p∂x------ 1 ε'2ξ2+( )

αρ------------------------- p

U2

ε------–+

∆tε

-----+⎩ ⎭⎨ ⎬⎧ ⎫

,=

C11ρ 1

ε--- uU

ε'ξpαρ----------+⎝ ⎠

⎛ ⎞ ∆t,–=

A22u 1

2--- ρu∆x

γpα----- 2γ 5 5γ– 6–( )ρu2– ∆t+

⎩ ⎭⎨ ⎬⎧ ⎫

,=

(15)

In expressions (15), the superscripts of the coeffi-cients on the left-hand sides indicate the variable towhich the coefficient is related.

In accordance with published results [7, 8], the cri-teria for the stability of the considered modification ofthe scheme of the particle-in-cell method are conditionsfor the positivity of the diagonal elements of the matri-ces A and B. A significant difference of expressions

(15) for the coefficients and from similarexpressions for these coefficients obtained according tothe explicit scheme of the Euler step of the particle-in-cell method is the following. The terms of the form p/αare involved in the right-hand sides of the expressions

B22u 1

2ε----- ρU∆x

1ε--- γpε'2ξ2

α----------------- ε'ξ U 2ε'ξu–( )[++

⎩⎨⎧

=

γ 1–( ) u 1 ε'2ξ2+( ) ε'ξU–( ) ]ρu– ∆t⎭⎬⎫

,

C22u 1

ε--- ε'ξγp

α-------------–

12--- ε'γ 5 2γ–( )u U+[+

⎩⎨⎧

=

+ γ 1–( ) U u–( ) γ 1–( ) 2γ 1–( ) U ε'ξu–( ) ]ρu+⎭⎬⎫

∆t,

A33v 1

2--- ρu∆x ρ u2 γ 1–( )v 2+[ ]∆t–{ },=

B33v 1

2--- ρU∆x

1

ε2---- γp

α-----+

⎩⎨⎧

=

---– ρv 0.5 U 2v+( ) γ 1–( ) 2γ 1–( )U+[ ] ∆t⎭⎬⎫

,

C33v ρv

ε------- 2U 4 γ–( )u+[–=

+ 3ε'ξ γ 1–( )v γ 1–( ) 2γ 1–( )u+ ]∆t,

A44E ρ u∆x

2---------- γ 1–( )E γ 2 γ 1–+( )u2+[ ]∆t+

⎩ ⎭⎨ ⎬⎧ ⎫

,=

B44E ρ

ε--- U∆x

2-----------

1ε--- γ 2 γ 1–+( )U2 1 ε'2ξ2+( )E+[ ]+

⎩ ⎭⎨ ⎬⎧ ⎫

,=

C44E 2ρUuγ 2∆t

ε--------------------------.=

Ammf Bmm

f

86


ENIKEEV

for and in the implicit and explicit schemeswith plus signs and minus signs, respectively. At lowsubsonic velocities, when α � 1, these terms make themain contribution to the right-hand sides of the expres-

sions for and . This significantly weakens theconstraints on the ratio ∆t/∆x at α � 1, which arecaused by the requirement of stability of the differencescheme. At α ≅ 1, the contribution of the terms of the

type is on the same order as that of ρu2. In this case,

the stability conditions for the implicit scheme are vir-tually the same as for the explicit one, because of whichthe implicit schemes are inexpedient to use at large α.Because of the nonlinearity of the initial set of Eqs. (1),the numerical algorithm has to be supplemented withan iterative process for calculating the pressure field.This significantly complicates the study of stabilitysince the iterative convergence essentially depends onthe geometry of the flow region. It is impossible to ana-lytically prove the convergence of the above iterativeprocess in complex-shaped regions; therefore, the con-vergence is determined experimentally in each specificproblem.

Note that, at small α, the terms of the type p/α cause

the right-hand sides of the coefficients , and to take positive values at any ratios ∆t/∆x. Some con-straints on ∆t/∆x are nonetheless imposed, since theconvergence of the iterative process may dependstrongly on this parameter. The practice of calculationsshowed that the implicit scheme is stable at ∆t/∆x = 0.1at α ≥ 0.001.

Calculation results. The test problem (Fig. 1) wasa problem of the flow of a gas–drop mixture in a con-vergent tube with the generatrix defined as

Calculations were carried out at = , / =

0.1, ρ20 = 0.001, d = 8 µm, and å0 being varied withinthe range 0.05–0.9. The accuracy was controlled bycomparing the numerical solutions obtained on differ-

ent computational grids corresponding to = and

= . The results differed within a range of 3–5%.

Since the flow had a low mass concentration of the dis-persed phase, the distributions of the parameters of thecarrier medium corresponded to the analogous distribu-tion of the parameters of a single-phase flow.

Ammf Bmm

f

Ammf Bmm

f

pα---

Ammf Bmm

f

F x( )2.2, x 0.5≤2.5 0.6x, 0.5 x 2.5≤<–

1, x 2.5.>⎩⎪⎨⎪⎧

=

∆x114------ ∆ t ∆x

∆x114------

∆x128------

The flow pattern in the convergent tube is shown in

Fig. 1. The level curves , where V = ,

that are obtained by the considered procedure (Fig. 1)and also published data [5] demonstrate the adequacyof the developed method to the physics of the problemconsidered.

The calculation of the two-dimensional isentropicflow of the gas from the convergent tube shows that thecurve for the flow rate as a function of the ratio of theoutlet pressure pout to the inlet pressure pin is in qualita-tive agreement with the well-known theoretical data [6]presented in Fig. 2. A certain difference consists in thefact that the theoretical data [6] were obtained for aone-dimensional flow, whereas the calculation was per-formed for a two-dimensional flow.

Va--- v 1

x( )2v 1

y( )2+

0.8

0 0.2

G/G*

Pout/Pin

12

0.6

0.4

0.2

0.4 0.6 0.8

3

Fig. 2. Two-dimensional isentropic gas flow from a conver-gent tube as (1) calculated in this work and (2, 3) deter-mined in the literature [6].

y/R

1

0 1 3 x/R1

2

2

1

1

Fig. 3. (1) Gas and (2) drop streamlines in an axisymmetricchannel.



Figure 3 presents the streamlines of the carrier phaseand the streamlines of drops in an axisymmetric chan-nel whose generatrices of the lateral surface consist oftwo parallel straight lines and circular arcs. At the chan-

nel inlet (x = 0), = = 0, M0 = 0.08, St = 6.4, andρ20 = 0.001. Figure 3 shows that, at a given St, all thedrops the radius of introduction of which in the initialsection varied from 0 to 1/2 fall onto the lower circulararc of the lateral surface of the channel. The distribu-

tions of and the reduced density of drops along thesymmetry axis (ξ = 0) and lateral surface (ξ = 1) of thechannel are presented in Fig. 4, which demonstrates

that the distribution of along the line ξ is typical ofthe flow of a pure (without particles) gas past a sphere.

Ahead of the sphere and behind it, has the minimalvalue, and on its stern, the maximal one. On the line

ξ = 1 at the points where the flow turns (x/R = 1.3), reaches the maximal value, and in the bottom region(x = 2), the minimal one. Analysis of the drop densitydistribution reveals that the concentration is maximal inthe stern part of the circular arc of the line ξ = 0 in thevicinity of the point x/R = 2. Note also that drops hardlyfall on the bottom part of the upper arc of the circle. Thedrop concentration distributions in different sections ofthe channel are shown in Fig. 5. It is interesting to notethat, in the section x/R = 3, the drop concentration dis-tribution is nonmonotonic. This is caused by the factthat, in moving past the lower arc of the circle, the dropswhose radius of escape from the initial section exceeds

v 1x

v 2y

v 1x

v 1x

v 1x

v 1x

0.5 interact with the gas and, thus, cause a significant

increase in , because of which the drop streamlinesare drawn together in the section x/R = 3.

Figure 6 presents the gas and drop streamlines in theflat channel modeling the geometry of a louver-typeinertial separator. The equations of the channel bound-aries have the following form:

the lower boundary yl,

the upper boundary yu,

Figure 6 shows that, for large drops at St ≥ 0.064 tosettle, a single corrugation is enough. With a decrease

v 2y

yl

1/6, 0 x 1/3≤ ≤1/6, π 3x 1–( )/2 1+sin[ ], 1/3 x 1≤ ≤1/6, x 3,>⎩

⎪⎨⎪⎧

=

yu yl 1/3.+=

U1x/U0

1.251.10

0.8

0.6

0.4

0.2

0 2

8

4

2

ρ2/ρ20

x/R

6

1

2

3

4

~~ ~~ ~~ ~~ ~~ ~~

Fig. 4. Distributions of (1, 2) the longitudinal component ofthe gas velocity and (3, 4) the drop concentration at ξ =(1, 3) 1 and (2, 4) 0.

4

01/3

ρ2/ρ20

r/R

1

2 3

2

2/3

Fig. 5. Drop distribution in different sections r of the chan-nel at x = (1) 2, (2) 1, and (3) 3.

y/R

0 1 3 x/R

3

2

1

1

2 4

Fig. 6. (1) Gas and (2, 3) drop streamlines in a separator atSt = (2) 6.4 and (3) 0.064.

88


ENIKEEV

in St, the necessary number of corrugations increases,and at St = 0.064, four corrugations are necessary for allthe drops to settle.

NOTATION

A—cross-sectional area of the channel;

a*—critical sound velocity in the gas;

c—specific heat at constant volume of the gas;

c2—specific heat of drops;

d—drop diameter;

Es—total energy of the sth phase;

es—internal energy of the sth phase;

F—friction force vector;

G—gas flow rate;

H—total enthalpy of the gas;

L—channel length;

lF—length of the zone of velocity relaxationbetween the gas and the drops;

lq—length of the zone of temperature relaxationbetween the gas and the drops;

p—pressure in the gas;

q—intensity of heat transfer between phases;

R—channel width;

r—channel radius;

S—gas entropy;

Ts—temperature of the Sth phase;

t—time;

U—velocity of the undisturbed flow;

u—longitudinal component of the gas velocity;

v—transverse component of the gas velocity;

—projections of the velocity vector on the coor-dinate axes;

vs—velocity vector of the sth phase;

x, y—coordinate axes;

β—parameter of thermal relaxation of the gas anddrops;

γ—adiabatic exponent of the gas;

λ—thermal conductivity of the gas;

µ—dynamic viscosity of the gas;

ρs—reduced viscosity of the sth phase;

—real density of drops;

ϕ—axis of the cylindrical coordinate system;

M—Mach number;

å0—Mach number of the undisturbed flow;

St—Stokes number.

SUBSCRIPTS AND SUPERSCRIPTS

i, j—discrete coordinates of a computational cell;

k—index of summation by coordinate axes;

n—number of a time layer;

s = 1, 2—number of a phase;

0—parameters of the undisturbed flow;

(1)—fixed cross section of the channel;

u∗—dimensionless and dimensional variables,respectively;

~—parameters at the Euler step of the particle-in-cell method.

REFERENCES

1. Soo, S.L., Fluid Dynamics of Multiphase Systems,Waltham, Mass.: Blaisdell, 1967. Translated under thetitle Gidrodinamika mnogofaznykh system, Moscow:Mir, 1971.

2. Nigmatulin, R.I., Osnovy mekhaniki geterogennykh sred(Principles of Mechanics of Heterogeneous Media),Moscow: Nauka, 1978.

3. Vasenin, I.M., Arkhipov, V.A., Butov, V.G., et al.,Gazovaya dinamika dvukhfaznykh techenii v soplakh(Gas Dynamics of Two-Phase Flows in Nozzles),Tomsk: Tomsk. Gos. Univ., 1986.

4. Rychkov, A.D., Matematicheskoe modelirovanie gazod-inamicheskikh protsessov v kanalakh i soplakh (Mathe-matical Modeling of Gas-Dynamic Processes in Chan-nels and Nozzles), Novosibirsk: Nauka, 1988.

5. Godunov, S.G., Zabrodin, A.V., Ivanov, M.Ya.,Kraiko, A.N., and Prokopov, G.P., Chislennoe resheniyemnogomernykh zadach gazovoi dinamiki (NumericalSolution of Multidimensional Problems of Gas Dynam-ics), Moscow: Nauka, 1976.

6. Loitsyanskii, L.G., Mekhanika zhidkosti i gaza (FluidMechanics), Moscow: Nauka, 1978.

7. Belotserkovskii, O.M. and Davydov, Yu.M., Metod krup-nykh chastits v gasovoi dinamike (Particle-in-CellMethod in Gas Dynamics), Moscow: Nauka, 1982.

8. Davydov, Yu.M., Differentsial’nye priblizheniya i pred-stavleniya raznostnykh skhem (Differential Approxima-tions and Representations of Difference Schemes), Mos-cow: Mosk. Fiz.-Tekh. Inst., 1981.

vsk

ρ20

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