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AIAA 2001-2649Development of Low-DiffusionFlux-Splitting Methods forDense Gas-Solid Flows

Jack R. EdwardsDeming MaoNorth Carolina State University,Raleigh, North Carolina

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Development of Low-Diffusion Flux-Splitting Methods for Dense Gas-SolidFlows

Jack R. Edwards *North Carolina State University, Raleigh, NC

Deming Mao ^North Carolina State University, Raleigh, NC

I. Abstract

The development of a class of low-diffusionupwinding methods for computing dense gas-solid flows is presented in this work. An artifi-cial compressibility / low-Mach preconditioningstrategy is developed for a hyperbolic two-phaseflow equation system consisting of separate solidsand gas momentum and continuity equations.The eigenvalues of this system are used to de-vise extensions of the AUSM+[1] and LDFSS[2]flux-splitting methods that provide high resolu-tion capturing of bubble growth and collapse ingas-solid fluidized beds. Applications to severalproblems in fluidization are presented.

II. Introduction

Fluidized bed reactors are prevalent in manyindustrial settings, and an accurate prediction oftheir response under different operating condi-tions has been a goal in multi-phase computa-tional fluid dynamics for many years [3-10] Pio-neering works, such as that of Gidaspow and co-workers [3, and references cited within], utilizedclassical pressure-based methods combined with

"Associate Professor, Campus Box 7910, NCSU,Raleigh,. NC 27695; [email protected]; (919) SIS-5264; Senior Member AIAA

* Research Assistant, Dept. of Mechanical andAerospace Engineering, Campus Box 7910, NCSU,Raleigh, NC 27695; [email protected]; (919) 515-5250;Student Member AIAA

Copyright © 2001 by the American Institute of Aero-nautics and Astronautics, Inc. All rights reserved.

somewhat diffusive upwind or hybrid schemesto compute many problems of technical inter-est. Sophisticated closures for solids viscosityand solids pressure (which acts to prevent solidscompaction to zero voidage) were later intro-duced [4,5], with a view toward improving pre-dictions with experimental data. Such data is in-variably three-dimensional, with wall effects andmulti-port injection of gas and solids all con-tributing to the gas-solid mixing patterns, butmany pioneering contributions idealized the pro-cess by assuming either two-dimensionality oraxisymmetry. Approaching the literature froma different perspective, one is left wonderingwhether such sophisticated closures are actuallynecessary if one utilizes high-resolution numeri-cal schemes on well-resolved, three-dimensionaldomains with proper treatment of geometricaleffects.

The present work is designed to progress to-ward this "large-eddy simulation" level of mod-eling of fluidized bed reactor physics. The firststep, presented in this paper, involves the de-velopment of high-resolution methods for thecomputation of dense gas/solid flows. The ap-proach differs from that in [6-10] in that the tech-niques are developed within an overall frame-work of time-derivative preconditioning, whichenables strong coupling of the governing equa-tions and a reconcilition of widely differing char-acteristic speeds within the system. Attentionis focused on the hyperbolic two-fluid model ofGidaspow [3], specialized for incompressible gas-solid flows and augmented by the addition of a


solids pressure model proposed by Boivin, et al.[10] With this basis, preconditioning techniquesfor solving the equation system for incompress-ible solids and gas phases are developed. The re-sulting eigenvalues are used to define extensionsof the AUSM+[1] and LDFSS [2] low-diffusionupwind schemes suitable for high-resolution sim-ulations of fluidized-bed phenomena. Applica-tions to classical model problems in fluidizationconclude the paper, and directions for furtherdevelopment are outlined.

III. Governing Equations

The developments outlined herein are basedon Gidaspow's hydrodynamics model B [3], agas-solids model that is rendered hyperbolic byconcentrating gas pressure effects solely withinthe gas phase and solids pressure effects withinthe solids phase. The governing equations arewritten as

dU dF dG n_____ I ___ I ___ __ Qdt dx dy (1)

where

U =

/nr _

ag

OigUg

a9V9 p =

asus

. asv3 .agvg

O-gVgUg

asvsasvsus

_ asv2s + ps IPs .

OLgUg

OigU2g +Pg/pg

OLgUgVg

asus '®su2

s +ps/ps0>SUSVS

5

0C(Ug - US)/pg

0

~C(Ug - Us) / pS

. (1 - pg/ps)asg - C(vg - vs)/ps .

s =

In this, ag is the gas-phase void fraction, as =1 — ag is the solids-phase void fraction, us, ug,vs, vg are the solids and gas-phase velocities, and

g is acceleration due to gravity. The gas-phaseintrinsic density pg and the solids-phase intrinsicdensity ps are assumed constant. The functionC scales the momentum exchange terms and isdefined as follows:

- V,\

and

C = 0.75CD-a.-2.65

dp

, ag < 0.8

(3)

.8 (4)

In the preceding expressions, /j, is the gas viscos-ity (assumed constant), dp is the particle diam-eter, and CD is the drag coefficient. The formfor the solids pressure is taken from Boivin, etal. [10]:

Ps = 4-a s

-)

(5)

where a55max is the solids volume fraction atmaximum compaction (taken as 0.64) and Cs isa scaling constant (taken as 0.02 for most calcu-lations). A solids "sound speed" can be definedthrough the relation

— PsCs(-OL*

(6)

The solids sound speed ranges from zero atzero solids voidage to infinity at maximum com-paction.

IV. Time-Derivative Preconditioning

To enable time evolution of this system, ar-tificial time derivatives of gas and solids pres-sure are added to the continuity and momentumequations. The modified equation system can beexpressed as

(7)dt dx dy


where the vector V — [pgiUg,vg,as,us,vs] and described above. The following choices for /3gthe preconditioning matrix P is and /3S are used in this work:

- 1Wj-2J2

Vg

000

0ag

0

000

00

<*g000

-1-Ug

~V9

-^fos+ ̂ es)

, PsQ-l a \+ -J*-Vs)

000

0as

0

000

00as

P =

vs(l(8)

In this, (3g is a reference velocity for the gasphase, Os is a scaling parameter, defined as 9S —•p- — \, and /3S is a reference velocity for ther s S

solids phase. The quantities p and p are refer-ence densites. Precise forms for all of these willbe defined later. The eigenvalues of P~~1A, whereA = -or/ are:

Me2ff) ± effj

' (9)where

_ ! |~

M? - | (H,

It is of note that the eigenvalues correspond-ing to the gas phase are identical with thoseobtained for Chorin's artifical compressibilitymethod, with an effective reference velocity ofA/-£-/32. The eigenvalues corresponding to thesolids phase are identical to those typically re-sulting from preconditioning matrices of theforms proposed by Turkel [11], Choi and Merkle[12], and Weiss and Smith [13], among others,with a specially-defined reference Mach numbergiven by Eq. (11).

The choice of the reference velocities /3g andPS and the reference densities p and p can haveprofound effects on the accuracy and efficiency ofschemes designed to solve the gas-solid system

/32 -

(12)

), (13)

with C/re£ and Uief s both set to user-specifiedconstants. The forms for /3g and /3S are con-sistant with the assumption of an incompress-ible gas phase and a "compressible" solids phase,in the sense that the solids phase can becomemore dense or dilute as conditions change. Thepreconditioning strategy for the solids phase re-places the solids sound speed by a quantity pro-portional to ps as the degree of compaction in-creases. In contrast, a "supersonic flow" situa-tion may prevail for dilute gas-solid flows, as thesolids phase velocity may be much higher thanthe solids sound speed.

The choice of reference densities p and p isless obvious. The selections p — pg and p — ps

render the eigenvalues independent of the densitybut result in excessively diffuse solutions thatdo not fluidize at the right conditions for theschemes discussed later. Better choices includep — p — pga.g + psas = pb, the bulk density, andp = pb^ p — ps. The latter choice renders thesolids-phase eigenvalues independent of density,while the former means that the bulk continuityequation evolves according to

L(L<*!>d

d

= 0 (14)

Both choices imply that numerical formulationsdiscussed next will be phase-coupled through theeigenvalues.

V. Flux-Splitting Schemes

For many two-phase and multi-phase equa-tion systems, determining the characteristicspeeds alone is a difficult task, even if the sys-tem is constructed so that real eigenvalues re-sult. Determining the associated eigenvectors isan even more time-consuming task, and schemes


that depend on those eigenvectors are likely to berather expensive on a per-iteration basis. Thiscomplexity increases if time-derivative precon-ditioning techniques need to be introduced toavoid excessive numerical stiffness. It may beof use, therefore, to consider schemes that donot depend on the details of the eigenstructureto a large extent. Examples of such schemesare the Lax-Friedrichs method, which is ratherdissipative, and such low-diffusion methods asAUSM+[1] and LDFSS [2].

Earlier works [1,2] have detailed the devel-opment of AUSM+ and LDFSS, both of whichcombine the the robustness and simplicity offlux-vector splitting methods with the accuracyof flux-difference splitting methods. Generalprocedures for extending these methods to low-Mach gas-phase flows have been presented in [14]and extensions suitable for real fluids undergo-ing equilibrium phase transitions have been pre-sented in [15]. AUSM-type methods for com-puting separated two-phase flows are known tobe under development by several groups [16,17],but as yet, little archival information is available.These techniques tend to differ quite widely intheir construction.

The present work presents extensions ofAUSM-h and LDFSS for the gas-solid system dis-cussed above. The formulation follows from thelow-Mach and real-fluid extensions of Refs. 14and 15.

The interface flux vector F in Eq. 7 is splitinto phasic contributions Fg -f Fs as follows:

F =

(15)Each phasic contribution Fk (k — gas or solids),evaluated at a cell interface, is then furthersplit into convective and pressure contributions

are discretized

separately in the following manner:

agug

agU2 +Pg/Pg

CtgUgVg

000

000

asusasu2

s+ps/psasusvs

1/2 1/2'

R

and0

Pk,l/2/Pk0

(16)

(17)

Differences between LDFSS and AUSM-f- relateto the chosen functional forms for Ujf and ^, 1/2-Both, however, are grounded in the concept ofa "numerical speed of sound" [18] that facil-itates the shift from a discretization suitablefor strongly compressible flows to one suitablefor incompressible flows. Two numerical soundspeeds, each associated with a particular phaseand developed from the acoustic eigenvalues, aredefined as follows:

(18)

Quantities appearing in Eqs. (18) and (19)are defined in Sections III and IV and arearithmetically-aver aged to the cell interface.Other quantities needed in the LDFSS andAUSM4- formulations are phasic Mach numbersat left (L) and right (R) states:

uk,

and polynomials in Mach number [1]:

M (2),*±I(M f c ±l) 2 , |M f c |< lMfi\ k otherwise,

±i(M f c±l)2±

(20)

(21)

(22)

(23)otherwise,

The numerals in the subscripts of M indicate thedegree of the polynomials.


TT+ —Uk ~

From this, Uj^r is defined for LDFSS as

) (24)

-,-) ' (25>where

A «->. i I A n^. I

-) (26)

M 1/2 -) (27)1,1/2

andApfc = Pfc.L ~Pk,R: (28)

The notation .M^ k L (for example) indicatesthe evaluation of the polynomial (22) using thephasic Mach number Mk at the left state. Thefunction M.\II is given as

(29)

if both (Mfc^l < 1 and \Mk,R < 1 and is zerootherwise. The function V2rr is equal to Pg^- forthe gas phase and M^aj (Eqns. (6) and (10))for the solids phase. As discussed in Refs. 2and 13 , the part of U^ that is proportional tothe pressure difference Ap/- (the "pressure dif-fusion" term) provides the necessary pressure-velocity coupling at very low speeds and facil-itates monotone resolution of non- grid aligneddiscontinuities at higher speeds. The particularform is a modification of that presented in Ref.2, following scaling arguments introduced in Ref.14.

For AUSM-f, U£ is defined as

= afc,i/2(max(°.

(30)

-M 1/2 -) (31)

where

and all other quantities are as defined above forLDFSS. The multiplication of the pressure dif-fusion term by 1 — M2rr serves to switch thisterm off whenever the local Mach number ex-ceeds unity. For the incompressible gas phase,the physical sound speed is assumed to approachinfinity, and M^ is set to zero.

The proper choice of pressure splitting is asubject of current debate. Earlier works [13,14]have shown that the original Van Leer / Liou" tyPe polynomial splittings lead to unphysicalsources of numerical diffusion for low-Mach flowsand can lead to scaling problems for real fluidflows. Modifications designed to prevent suchadverse behavior have been proposed in Refs. 13and 14 but all leave something to be desired. Inthis work, a more complete form for a generalpressure splitting is presented and tested.

As a starting point, a general Van Leer /Liou - type polynomial splitting is defined forphase k as

where ra — 1, 3, or 5 corresponds to polynomialsof different degrees, defined as follows:

(34)otherwise,

/ /Q \ r, ——

(5),fc

otherwise,(35)

\Mk\ < 1otherwise,

(36)Eq. (33) can be rewritten without approxima-tion as

1,^0. ),k,R^(Pk,L - Pk,R)

•z(pk,L + Pk,R)(P~(,(37)

which can be further separated into cell-averageplus diffusive components:

(32) Pfc,i/21,

= 2(Pk,L+Pk,R)


£<1 -Z— A U

- Un] + Rn+l'm), (40)

(38)

The third term on the right-hand side of (38) isthe major problem, in that when the polynomi-als are evaluated using the phasic Mach numbers,the term may be excessively large for low-Machflows. A better scaling, one also valid for real flu-ids, is found by replacing \(pk,L + Pk,R) in

term by pk&k i/2' ^e final form of the new pres-sure splitting is thus

1Pk,l/2 = ^(Pk:L+Pk

!,-,+(m),k,L+ ' Pk,R)

+

(39)

This modification ensures that the second dif-fusive term will scale as the velocity magnitudeas the Mach number becomes small. The newsplitting represents an improvement over the lin-earized splittings presented in Ref. 14 in that theproper response at sonic transitions is ensured.One also can replace the cell-average pressureby 1/2 - At low Mach numbers, this will be

re~less dissipative than the choice Pk&\ 1/2-suits presented in this paper use Eq. (39) exclu-sively, but there are indications that the choice

may be better overall.

The current implementation for LDFSS usesthe first-degree polynomials P,^ k exclusively.The AUSM-h implementation tests all threepolynomial forms.

VI. Time Integration

The schemes outlined in the previous sectionhave been implemented into a code for solvingtime-dependent, two-dimensional or axisymmet-ric gas-solid flows. A dual time-stepping methodis used to solve the following semi-discrete rep-resentation of Eq. 7:

p ay j_a^ dR_dw'

In this, R is the steady-state residual, J^ is theassociated Jacobian matrix, m is a subiterationindex, n is the physical time index, Ar is a lo-cal time step, At is the physical time step, andW — \pg, a.gUg, OLgVg, a.s , asus, a.svs}. This choiceof solution vector was found to be more stablethan V given above.

Jacobian matrix entries corresponding tothe convective terms in R are approximated asfollows:

W,t+1/2

(42)with

A —max|A5 , max |A5|,max |A5|] (43)

This can be seen as a linearization of a mod-ified Lax-Friedrichs flux for the preconditionedsystem, constructed so that the gas- and solids-phase components are each scaled by the corre-sponding maximum (in magnitude) eigenvalue.While not ideal, this approach provides a robust"driver" for the dual-time stepping algorithm.

VII. Results

In this section, the methods outlined ear-lier are tested for two problems in fluidization -compaction of a solids bed and bubble formationwithin a minimally-fluidized bed. In both cases,a rectangular 0.39m x 0.58m domain contain-ing 65x65 mesh points is used. Nominal initialconditions are as follows:

0 < y < hbed, ag = 0.440 < x < 0.39 as - 0.56

Ug —— 0

agVg = 0.28 m/sus - 0vs - 0pg = Poo + psasg(hbed - y)


0 < x < 0.39oig = 0.999as = 0.001

ctgVg — 0.28 m/sus = 0vs = 0Pg = Poo

Other necessary parameters are chosen as fol-lows: dp (particle diameter) — 500 /^m, ps —2660 kg/m3, pg = 1.235 kg/m3, /* - 20 x 10~6

kg/(m-s), poo = 101000 Pa. For these condi-tions, a minimum fluidization velocity of umf —0.64 m/s is obtained [3], as is a minimum super-ficial fluidization velocity of agum£ — 0.28 m/s.Nominal boundary conditions are as follows:

0 < y < 0.58,x = 0,z = 0.39

as extrapolated from interiorpg extrapolated from interiorU = 0Vg extrapolated from interiorus = 0vs extrapolated from interior

0 < x < 0.39, as extrapolated from interiory = 0 pg extrapolated from interior

ug = 0OL9V9 ~ 0.28 m/sus extrapolated from interiorvs = 0

0 < x < 0.39, as extrapolated from interiory = 0.58 Pg=Poo

ug extrapolated from interiorvg extrapolated from interiorus extrapolated from interiorvs = 0

These boundary conditions are applied at a cellinterface using an array of ghost cells placed out-side the physical domain and standard finite-volume strategies. The exception is the gas pres-sure at the inflow (y = 0) boundary. Here, itis necessary to linearly extrapolate the gas pres-sure to the ghost cell so as to preserve the properpressure-height relation.

The schemes are extended to second-orderaccuracy using standard slope-limiting proce-dures applied to the primitive variable vectorV. Both Van Leer and Superbee limiters are

used in the results that follow. The simulationsare advanced in time using a fixed time stepof 1 x 10~4 seconds, with 25 subiterations per-formed per time step. Larger numbers of subit-erations changed the results only minimally. Asthe focus of this work is toward the initial assess-ment of a new approach for simulating problemsin fluidization, only comparisons illustrating theeffects of different algorithmic parameters andmethods are shown. A more detailed assessmentversus relevant experimental data is in progress.

Attention is focused primarily on theLDFSS implementation, with baseline parame-ters of Uiefig - UieftS = 5 m/s, Cs - 0.02,p — p — pb. The baseline implementation alsouses the P,^ k polynomials and the Van Leer lim-iter. Selected calculations employ the AUSM-fextension.

Figure 1 displays results from a simulationof solids compaction in the bed using the base-line LDFSS implementation with differing val-ues of Cs. For this simulation, ^bec[ is taken

1

0.9

0.8

0.7

0.6

S"0.5

0.4

0.3

0.2

0.1

°c

initial profile

_._._._ c,= 0.1— — — - Cs=0.02

-

- - - - - - ~"~~"""-~"^

r " ~ ~ - - ̂ .

!•j i

|ji

\\, , , , i , , , , i , , , i i , , i I

) 0.1 0.2 0.3 CX(m)

\

\

\

t

I

\

.4 0.5

Figure 1: Final voidage pattern for simulationof bed compaction

as 0.41 m, and the inflow superficial velocity isset to zero. The initial solids distribution thusmoves downward under the influence of gravity


until a balance between gas pressure, solids pres-sure, and gravitational force is achieved. Thevoidage pattern shown at the end of the sim-ulation (15000 iterations) is nominally steady,though some low-frequency oscillations still per-sist. The figure indicates that near maximumcompaction is achieved at the bottom of the bedfor low values of Cs. Decreases in Cs result inthe sharpening of the final voidage profile.

The remainder of this paper discusses theperformance of the developed techniques for theproblem of jet-induced bubble formation in aminimally fluidized bed with /i^ed eclual to 0-29m. This case has been studied by several au-thors under different conditions [6, and refer-ences cited within, 19]. In the present work, atwo-dimensional jet of width 1.5 cm and super-ficial velocity 5.2 m/s is centrally located at thebottom of the bed and is introduced into theminimally-fluidized bed at time t = 0. The jetinduces the formation of several bubbles, whichrise, deform, and collapse after reaching the sur-face. The results that follow present snapshotsof the solids voidage at times t = 0.3 s, t — 0.54s, and t = 0.85 s.

The effect of modifying UTef and J/ref 5 onthe baseline LDFSS implementation is shown inFigure 2. The reference velocities are set equalto one another and are varied from the baselinevalue of 5.0 m/s as shown. The reference solu-tion is initially symmetric, but develops an asym-metry around t = 0.54 s which becomes morepronounced during the bubble collapse / solidsspouting stage at t — 0.85 s. As the referencevelocity is decreased, the computed bubbles be-come more rounded and are generally larger, andthe solution retains its symmetry for longer pe-riods of time.

Figure 3 illustrates the effect of the choice ofCs on the baseline LDFSS solutions. This con-stant scales the solids sound speed, and highervalues imply that the solids phase will respondin a more elliptic manner. In this figure andthe next two, only the contour corresponding toa median "bubble edge" of as = 0.3 is shownfor reasons of clarity and brevity. As shown,

t=0.3 s t=0.54 s t=0.85 s

0.5 m/s

2.0 m/s

5.0 m/s

Figure 2: Effect of reference velocity onfluidized bed response

the choice of Cs — 0.1 results in minimal bub-ble growth. The lowering of Cs from the baselinevalue of 0.02 to 0.01 results in slightly larger bub-bles and a more symmetric breakdown pattern.

The effect of the spatial accuracy of theLDFSS solution is shown in Figure 4. The first-order solution maintains symmetry, but the gen-erated bubbles are overly pointed at the cen-terline. As discussed in Ref. 6, this responseis typical of first-order upwind methods for thisproblem and can be traced to excessive numer-ical diffusion in the solids momentum equation.Still, the first-order results presented here arenoticeably better than the first-order solutionsof Ref. 6. The extension to second-order accu-racy results in more rounded bubbles and a rapidbreakdown to a non-symmetric, rather chaotic


t=0.3 s t=0.54 s t=0.85 s

< ? w < y0

C =0.01

C =0.1

t=0.3 s t=0.54 s t=0.85 s

C =0.02

0

Superbee

Van Leer

1 st order

Figure 3: Effect of Cs on fluidized bed response Figure 4: Effect of spatial accuracy on fluidizedbed response

bubbling bed. The use of the Superbee limiterseems to accelerate the breakdown of the sym-metric structure and results in generally morerounded bubbles.

Figure 5 illustrates the effect of the choiceof the reference densities p and p on the fluidizedbed response. As discussed earlier, the choices ofp — pg and p — ps (Case A) render the eigenval-ues independent of the density. This predictionis different from the others in that the bed ex-pands more with gas injection. The predictedbubble shapes are more rounded, and the so-lution maintains its symmetry reasonably well.The choices of p — pb and p — pg (Case B) resultin predictions very similar to the baseline case.While the results for Case A appear promisingat first glance, these choices do not respond aswell in other tests. Figure 6 displays results froma simulation in which the inflow superficial ve-

locity is increased to two times the minimumfluidization velocity in the absence of the cen-terline jet. The expected result is a transientexpansion of the fluidized bed to a new position.As predicted by the baseline formulation, thisexpansion is accompanied by transient bubbleformation and collapse, and the position of thegas-solid interface eventually oscillates about Y— 0.4 m. Nevertheless, solids mass conservationis excellent (less than 0.2% mass loss / gain). Incontrast, the choices of Case A result in an un-physical inversion of the bed, with a solids massloss of 28%. The reasons for the failure of themore natural choices of p — pg and p — ps to pro-vide a proper response are not completely clear,but it is noteworthy that the gas-phase pressurediffusion terms in Eqs. 26 and 27 are nearly fiftytimes larger for the Case A choices than for the


t=0.3 s t=0.54 s t=0.85 s

0°0

<?

Case A

Case B

baseline

Figure 5: Effect of reference density choice onfluidized bed response (Case A: p — pg, p — p5,

Case B: p = pb, p = ps)

1

0.9

0.8

0.7

0.6

s"o.5

0.4

0.3

0.2

0.1

n

: ——————— |n|tia| profile

- — — — - baseline reference density (15000 Iterations)I _._. — ._ Case A reference density (15000 iterations)

7

r

r /i i *

^ / /ft / "i' > '/' i: ' ' " • ' ' / 'L i / •' >'./ i; \f: / 1L •' , !

1

L / i: .'' !"i"^ i i 1 i IL-I — r -4- -T- i — i--*-

^ / * "7

1 / 1 /

\ / i !v 1 I

i !i ii 'i ;i /

1— » -H • -t- • r-U — r • T "i~*r"T" i i i

Y(m)

Figure 6: Effect of reference density choice on1-D fluidized bed response (Case A: p = pg,

p = ps, Case B: p = p6, p - ps)

predictions are minimal, though some evidenceof increased numerical diffusion with the use ofthe higher-degree splittings is present in the formof slightly reduced bubble sizes.

VIII. Conclusions

baseline values.

Some results from the AUSM+ extensionare shown in Figure 7 for the same parameters asused in the baseline LDFSS scheme. ComparingFigure 7 with Figure 2, it can be seen that thepredictions of AUSM+ are generally similar tothose of LDFSS, though AUSM+ seems to cap-ture some fine scale features more sharply. Onereason for this may be the inclusion of an ad-ditional diffusion mechanism, scaled by the ab-solute value of the interface pressure difference,in LDFSS. As discussed in Ref. 2, this termacts to suppress the "carbuncle" phenomenon of-ten noted in supersonic blunt-body flows, butits utility in very low-speed calculations is ques-tionable. Figure 6 also shows that the effects ofhigher-degree pressure splittings on the AUSM+

A new approach for simulating dense gas-solid flows has been presented. The approachcombines an artificial compressibility / precon-ditioning strategy for a particular two-phaseflow model with extensions of the LDFSS andAUSM+ upwinding techniques. Simulations ofbubble formation within a minimally fluidizedbed have been conducted to illustrate differentfacets of the developed schemes. It is shown thatthe choice of reference velocity and the scalingof the solids sound speed both have marked ef-fects on the ability of the schemes to capturethe response of the bubbly bed properly. Val-ues of the reference velocity of the order of theminimum fluidization velocity seem to promotephysically consistent bubble shapes, as do lowervalues of the solids sound speed scaling factor.As expected, higher-order simulations using the

10


t=0.3 s t=0.54 s t=0.85 s

5th degree

3rd degree

1 st degree

Figure 7: Effect of pressure splitting onfluidized bed response (AUSM+)

Van Leer and Superbee limiters provide a muchmore physically consistent response, and some-what surprisingly, the choice of reference den-sity also impacts the quality of the simulations.Differences between LDFSS and AUSM-f- predic-tions are found to be relatively minor.

While somewhat specialized to a particu-lar application in this work, many of the ideaspresented herein should be directly extendableto other applications of low-diffusion upwind-ing techniques. More specifically, a new formof the pressure flux splitting has been developedthat overcomes many of the scaling and con-sistency problems of earlier efforts, more self-consistent forms for adding pressure diffusion tothe mass flux have been developed, and the con-cept of a "numerical sound speed" in enablinggeneral low-Mach calculations has been extended

to two-phase flows. These developments bringus one step closer to a unified, well-structuredflux-splitting approach capable of handling gas-dynamic, real-fluid, and multi-phase flows withequal ease.

IX. Acknowledgments

This work has been sponsored by the U.S.Environmental Protection Agency under cooper-ative agreement CR-82795701-0. IBM SP-2 com-puter time has been obtained through a grantfrom the North Carolina Supercomputing Cen-ter.

X. References

[1] Liou, M.S., "Progress Towards an Im-proved CFD Method: AUSM+," AIAA Paper95-1701, June, 1995.

[2] Edwards, J.R. "A Low-Diffusion Flux-Splitting Scheme for Navier-Stokes Calcula-tions," Computers & Fluids, Vol. 26, No. 6,1997, pp. 635-657.

[3] Gidaspow, D. Multiphase Flow andFluidization Academic Press, Boston, 1994.

[4] Sinclair, J.L. and Jackson, R. "Gas-Particle Flow in a Vertical Pipe with Particle-Particle Interactions," AIChE Journal, Vol. 35,1989. p. 1473.

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