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AD-Ai49 313 AVERAGING EFFECTS IN MODELS OF THREE-DIMENSIONAL i/iTWO-PHASE FLOWS(U) ARMY BALLISTIC RESEARCH LAB ABERDEEN N
U C PROVING GROUND MD A K CELMINS OCT 84 BRL-TR-2598
INLASSIFIED 501-AD-F300 508 F/G 19/4 N
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MICROCOPY RESOLUTION TEST CHARTNA NA AL UP[ I -I , ANt At
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AD-A 149 313 ____
B ADB9R TECHNICAL REPORT BRL-TR-2598
L .
AVERAGING EFFECTS IN MODELS OFTHREE-DIMENSIONAL TWO-PHASE FLOWS
Aivars K.R. Celmins
October 1984
APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNUMITED.
US ARMY BALLISTIC RESEARCH LABORATORYABERDEEN PROVING GROUND, MARYLAND
- 84 11 07 00,
Destroy this report when it is no longer needed.Do not return it to the originator.
Additional copies of this report may be obtainedfrom the National Technical Information Service,U. S. Department of Commerce, Springfield, Virginia22161.
The findings in this report are not to be construed as an officialDepartment of the Army position, unless so designated by otherauthorized documents.
The use of trade names or manufacturers, names in this reportdoes not constitute indorsement of any commercial product.
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TcchnicaI Report BRL-TR-2598
4. TITLE 5and Subtle) S. TYPE OF REPORT & PERIOD COVEREDAverag ng Ettects in Models of Three-DimensionalTwo-Phase Flows
6. PERFORMING ORG. REPORT NUMBER
AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(S)1ivars K. R. Celmins
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKLIS Army Ballistic Research Laboratory AREA & WORK UNIT NUMBERS
ATTN: ,MXBR-IBDAberdeen Proving Ground, Maryland 21005-5066
I I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATEOS Army Ballistic Research Laboratory October 1984ATTN: A LXBR-OD-ST 13. NUMBER OF PAGESAberdeen Proving Ground, Maryland 21005-5066 8214 MONITORING AGENCY NAME & ADDRESS(If different from Controlling Office) 15. SECURITY CLASS. (of this report)
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16. DISTRIBUTION STATEMENT (of this Report)
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18 SUPPLEMENTARY NOTES
'9 KEY WORDS (Continue on reverse side If necessary and identify by block number)Voltime Average Interior Ballistics Calculations Lattice ArrangementsCas-Parrtcle Mixture Undulations of Flow-Undulations Mean Particle DistanceTwo-Phase Flow Particle Induced Undulations Neighbor DistanceCas Volume Fraction Bounds of Undulations Smoothing by Averaging
L'irt ikcl Aggregate Boundary20 ABS"ACT rCCourtfue m reverse efi of tmac, e m s Idenwify by block number) ,.
In ,rder to avoid the treatment of individual particle motion In two-phas(- flow do-scrtption, one can use volume averaged descriptors. However,if the averaging volume is too small then individual particles can causelarge undulations of the averages. In this report the amplitude of suchlindulations is estimated by a bound. It is shown that in general, one mustavI.rage over 30-150 particles in order to obtain reasonably smooth averages.A consequence for interior ballistics calculations is that volume averages
WO F OR;"11JO 147n EDITION OF I NOV 6S IS OBSOLETE UNCIASSIFIIEI)
SECURITY CLASSIFICATION OF THIS PAGE (11Ne.m Dea Ente,.d)
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UNCLASS IF IEDSECURITY CLASSIFICATION OF THIS PAOE(Vhan Data 3ntee
20. ABSTRACT (Continued)
may be used to model the core flow but in general they cannot be used toresolve radial structures in two phases. Also presented are examples of therepresentation of a gas-particle aggregate boundary in terms of the gas
volume fraction, and an approximate expression is derived for the transitioncurve.
*!
UNCLASSI FI)EDSECURITY CLASSIFICATION OF THIS PAGE(Wten Date Entered)
TABLE OF OCIITEMT
PageTA.BLE OF 00TNTS.........................................................3
LIST OF FIGURES ......... o....................................... 5
1. INTRODUCTION ................................ ........ o....o...... 7
2. PRINCIPAL RESULTS ...... o........................................8
3. GAS VOIJJM FRACTION AS A FUNCTION (F ME AVERAINGVOLlIE .... .. 13
4 . GAS VOLUME~ FRACTIONI PROFIL.ES............ ............................... 34
5 . DISCUSSION OF ME! RESULTS ................................ ......... .36
A.PPEND)IX A. LATTICES ................o..................o.....o..45
APPENDIX B. NEIGHBOR DISTANCE AND MfAN DISTANCE .............. 55
APPENDIX C. ALGORITHM FOR GAS VOLUME FRACTION CALCULATION .... 61
APPENDIX D. GAS VOL.U!E FRACTION AT A PARTICLE AGGREGATE
BOUNDARY............................................ 65
APPENDIX E. UNDULATIONS OF AVERAGE FUNCTIONS................... 69
DISTRIBUTION LIST.....................................................o..77
3
.6
%- %71. %-. -7
k-L- 7 - -
1 Averaging Sphere Radius for Given Tolerance I0Ito, ......... S
2 Minimum Number of Particles in -n Averaging Volumefor given Tolerance Jl and k = 0.. ...............12
3 Gd Volume Fraction Dependence on Averaging Radiils.
Square Cylinder Lattice, I = 0.5 ............................ 16
4 ai Volume Fraction Dependecico on Averaging Radius.Sq ire Cylinder Lattice, -t = 0.9 ............................ 17
Gas Volume Fraction Dependence on Averaging Radius.triangular Cylinder Lattice, i= 0.9........................ 18 S
6 Gas Volume Fraction Dependence on Averaging Radius.Leap-Frog Square Lattice, n% 0.9 ......................... 19
7 Gas Volume Fraction Dependence_on Xveraging Radius.Leap-Frog Triangular Lattice, (t= 0.9 ....................... 20 0
Extremes of Gas Volume Fraction as Functions of AveragingRadius. Square Cylinder Lattice,.= 0.5................. 21"
9 Extremes of Gas Volume Fraction as Functions of AveragingRadius. Triangular Cylinder Lattice, q= 0.5 ................ 22 0
I0 Extremes of Gas Volume Fraction as Functions of AveragingRadius. Leap-Frog Square lattice, %= 0.5 ................... 23
11 Extremes of Gas Volume Fraction as Functions of AveragingRadius. Leap-Frog Triangular lattice, .= 0.5 ............... 24 0
12 Extremes of Gas Volume Fraction as Functions of AveragingRadius. Square Cylinder attice, i0. 9 .................... 25
13 Extremes of Gas Volume Fraction as Functions of AveragingRadius. Triangular Cylinder Lattice, a= 0.9 ............... 26
14 Extremes of Gas Volume Fraction as Functions of AveragingRadilis. Leap-Frog Square attice, -f 0.9 ................ 27 .
15 Extremes of Gas Volume Fraction as Functions of AveragingRadius. Leap-Frog Triangular Lattice, = 0.9 ............... 28 •
16 Extreme Deviations of Gas Values Fraction.Leap-Frog Triangular attice, 0.9 ........................29
17 Extreme Deviations of Gas Volume Fraction.Leap-Frog Triangular Lattice, &- 0.9 ........................ 30
5
LIST ( FIGURES (Continued)Figures Page
18 Extreme Deviations of Gas Volume Fraction for AllLattices. a = 0.5 ........................................ 31
19 Extreme Deviation of Gas Volume Fraction for All
lattices. a = 0.9 ................ .......... .......... 32
20 Gas Volume Fraction at a Particle Aggregate
Boundary. a = 0.5 ........................................ 35
21 Gas Volume Fraction at a Particle Aggregate
Boundary. a - 0.7 ... ...... . ...... ....... 37
22 Gas Volume Fraction at a Particle AggregateBoundary. a = 0.9 .......................................... 38
23 Extreme Values and Estimated Bounds of Gas Volume Fraction at a
Particle Aggregate Boundary. a = 0.5, R/Lm m 1............. 39
24 Extreme Values and Estimated Bounds of Gas Volume Fractionat a Particle Aggregate Boundary. a = 0.9, R/L - I ......... 40
25 Extreme Values and Estimated Bounds of Gas Volume Fractionat a Particle Aggregate Boundary. a= 0.9, R/ l = 2 .......... 41
A.I. Square Cylinder Lattice ........................... 48
A.2. Triangular Cylinder lattice................................. 48
A-3. Leap-Frog Square lattice ............................... 50
A.4. Leap-Frog Triangular lattice ................... 50
C.1. Intersection of Spheres ..................................... 63
7
6
... ...0.. '
p
1. IUlRODIUCT ION
A common method for the derivation of a manageable flow description andof governing equations for flows of gas-particle mixtures is the averagingof local flow properties over a volume.* The averaging produces from theheterogeneous local flow properties smooth flow parameter functions whichprovide representative descriptions of average properties of the particle
aggregate and of the gas between particles. The smoothing is most effectiveif the averaging volume is so large that the contributions of singleparticles to the average is negligible. Therefore, it is reasonable to
choose a large averaging volume. On the other hand, any volume averagingsmoothes, reduces and distorts flow structures particularly those which havean extension smaller than the averaging volume. Hence, in order not to loseflow structures of interest one should choose a small averaging volume. Inorder to make, under these conditions, a rational choice of the size of theaveraging volume one needs a quantitation of the smoothing effect of volumeaveraging.
In this report a quantitation of the smoothing effect is obtained froman investigation how undulations of the gas volume fraction functiona depends on the size and distribution of particles and on the size of theaveraging volume. Other flow parameters can be shown to have undulationsthat are proportional to the undulations of a. The result of the
investigation is an estimate of bounds for the undulations of a in terms ofthe averaging parameters. Using this estimate one can choose a tolerance1ovel for the undulations and obtain a corresponding minimum size of theaveraging volume. Flow structures with extensions smaller than the chosenaveraging volume are distorted and reduced by the averaging, that is, theyare not correctly represented by the average functions.
We illustrate the smoothing of flow structures by considering a plane
boundary of a region with uniform particle distribution. In concept, such a
boundary is a narrow transition zone between the regions with a = 1 (gasonly) and a = a (the average gas volume fraction in the mixture region).The a obtained by volume averaging has instead a relatively wide transitionzone which is spread out over a diameter length of the averaging volume.The width of the transition zone cannot be reduced arbitrarily withoutpenalty, that is, without increasing indulations of the average flow
parameters. Hence one has to choose between a smooth representation of theaverage flow field and an accurate representation of the boundary of the
region. The estimated bounds of the undulations can help one to make thechoice rationally.
*Cetimns, AVvau K.R. and Schmmitt, James A., Thtee-dimenzionao modeLing o6
gaz-cumbtLtng softd two-phaze 6fows, Proceedings o6 the Third Multi-phazeFCuw and Heat Transfej Spnposium-Workshop, pp. 681-698, Ed., T.N. Vezitogtu,l1-20 Apt 1983, Miami Beach, FL.
7
* P
In Section 2 we present the principal results of the investigation: aquantitative estimate of bounds for particle induced undulations of averageflow parameters. Particulars about the derivation of the estimate are givenin Section 3. Section 4 provides numerical examples of the representationof a particle aggregate boundary in terms of a. The transition profile canbe computed by an approximate formula which is also given in Section 4. Allcalculations are for a spherical averaging volume and spherical particlesarranged in regular three-dimensional lattices. However, by consideringdifferent types of lattices we have obtained results that are representativefor all regular particle arrangements. Also, the restriction to sphericalparticles is of little consequence if the particles are small compared tothe averaging volume. Section 5 contains a discussion of the results.
2. PRINCIPAL RESULTS
In Section 3 we consider L1 P gas volume fraction a within an averagingsphere with the radius R, and investigate undulations of a due to thelocation of the sphere. Let the particles be arranged in a regular three-dimensional array, let the particle radius be s, and let a be the limit of
0 the gas volume fraction as the averaging sphere becomes infinite. We definea mean distance Lm between the particle centers by
Lm = 2s(l - c) -1/3 (2.1)
A motivation for this definition is given in Appendix B, where it is alsoshown that Lm is 10-24% larger than the smallest distance between particlecenters. If the averaging sphere is finite, then one obtains instead of a avalue of the gas volume fraction that depends on the position of thesphere. Let Aai be the difference between an actual value of a and the limitvalue a:
Aa=a-ci . (2.2)
Ai generally depends on the position of the sphere as well as on a andLm/R. In Section 3 we show that its magnitude is bounded by
*A < 0.5 (1-a) a2 (L,/R)2 (2.3)
independently of the position of the averaging sphere. Eq. (2.3) is basedon sample calculations with different lattices for I ' R/Lm e 4 and 0.5a < 0.9. Because the formula also produces the correct limit Aa - 0 for
a 1, it can be used as an estimate for all a )0.5. Extrapolation to
SB
smaller values (a < 0.5) should be done with reservations, and the sameapplies to, extrapolations to R/Lt > 4. The domain R/L. < 1 is of little
practical interest because there the undulations are too large.
Eq. (2.3) is graphically displayed in Figure 1 as a relation between a
maximum tolerance Ito, and the corresponding R/L. which guarantees that
the undulations are less than the tolerance. The three curves are shown for
a = 0.5, 0.67 and 0.9, respectively. The value a = 0.67 is according to Eq.(2.3) the worst case, that Is, in this case one needs the largest R/ nj inorder to suppress the undulations. The solid lines indicate the domain for
which the formula (2.5) was developed and tested by sample calculations.Extrapolations are indicated by dashed lines. Using Figure 1 one obtains,
for instance, that the undulations IAaI are smaller than 0.01 for R/Lm > 2.0
in cise x = 0.9, and for R/L. > 2.5 in case a = 0.5.
The relation (2.3) also can be expressed in terms of the number N of
particles in the averaging volume. To arrive at such an expression we use
the approximation
N = (2R/lm)3 (2.4)
(see Eq. (3.5)). Substituting this approximation into Eq. (2.3) one obtains
Hf K 2.0 (1 - a) 2 N-2 3 (2.5)
For spherical averaging volumes Eqs. (2.3) and (2.5) are equivalent.
However, because the latter also can be used for non-spherical averaging
volumes,we shall restrict further discussions to Eq. (2.5). let IA-Itol be
a tolerance for the undulations. Then one can reformulate Eq. (2.5) as the
following condition for N
4 N > [2 (I - a )j3/2 3 IAaX-3/ 2 (2.6)~tol
If N satisfies Eq. (2.6) then IAaf a <l ltolIf a is close to one, then it is reasonable to replace the fixed
tolerance by a tolerance level that is a fraction of I -a, for instance, by
the condition
A K k (I - ) (2.7)
9
10-8
6-%
R/Lm
2-
0.8 -
0.6-
0.40.001 0.002 0.004 0.01 0:02 0:04 0.1
jAo 1to1
Figure 1. Averaging Sphere Radius for Given Tolerance IAaL to
10
with a proper k 1 1. The corresponding condition on N is
(i/ k) 32 (2.8)
One miy combine both conditions on N by requiring that N should be larger
than the maximum of the expressions (2.6) and (2.8). Figure 2 shows the
res it for k = 0.2 and various values -f a . According to that figure, one
needs 32-150 particles in order to keep the undulations below the tolerance
levels !A\Itol = 0.01 and k = 0.2 within the range 0.7 4 a < 1.0. On thether hand, if one is content with the tolerance level IAItol = 0.03 and k
=.2 then 32 particles suffice to represent reasonable average properties.
One cAn obtain more convenient and less sharp formulas than Eqs. (2.6)
And (28) by choosing the worst case conditions, namely, a =2/3 for Eq.
(. a and = 1.0 for Eq. (2.8). Then the combined condition on N is
N > ma._ ~2 9/2 A1-3/2 (2/k) 31/2ma X 3) 1tol 2/k(2.9)
Th,, First argument on the right hand side of Eq. (2.9) corresponds to the
rihL~nost curve in Figure 2. The undulations of averages of other flow
parim4t-,rs, such as gas density, are closely related to the undulations of
1. td, 1nve ;tigate this relation in Appendix E and show by examples that the
lmpiitude A, of particle induced undulations of an average gas property 1 isproortional to 5 . If the average has a gradient and changes by 6p along a
di ta-ce equal to the diameter of the averaging sphere, then
I I j5f Aa /3ai) .(2.10)
If n= but there are local particle induced inhomogeneities, then the
1ImIII itions again are proportional to Aa and to the amplitude of the local
I rhnm,)Feneitie. In summary, undulations of a usually induce proportional
!nd,+i! tions of other flow parameter averages.
some consequences of these findings for interior ballistics
computations are discussed in Appendix B, where it is shown that the mean
distance T between particle centers typically is as large as three initialprnpollant particle diameters. According to Figure 1, the corresponding
r-idii' R of the averaging sphere should be larger than six to eight particle !
liaieters (assuming IAltol = 0.01). This means that a two-phase Theory
which Is based on averaging may not provide correct information about radial
fliw -tructures in interior ballistics because the minimum diameter of the
averaglng sphere is approximately equal to the diameter of the barrel. Such
11
.... ... . . . . . .. ., . .f .. . .: . ......i 7. ..... ... ... ... .. ......", .-... .•.. ..
*- 7
1000800\600 -
400 =.6
0.9
N 20.0.950.97
6 100-80-
600.99
40-32
20-
10-0.001 0.002 0.004 0.01 0:02 0.04 0.1
S Figure 2. Minimum Number of Particles in an Averaging Volumefor given Tolerance IA&QTtol and k -0.2
12
a theory still can be used to represent averages over cross-sectional
segments of the gun tube, that is, it can be used to calculate the interior
ballistics core flow.
Volume averaging smoothes and distorts not only the undesirable
heterogeneities -aased by single particles, but also other flow structures,
particularly if they have extensions less than a diameter of the averaging
volume. An example of a flow structure with a small extension is the
boundary of a particle aggregate. Let the average gas volume fraction in
the aggregate be a and let the aggregate occupy the half-space z > 0.
Conceptually one thinks of the boundary at z = 0 as a narrow transition
between a region with a = 1 (gas only) and a region with a = a (gas particle
mixture). The correctly defined a has, instead a finite transition zone
between a = 3 and a = a with a width equal to a diameter of the averaging
volume. If the volume is a sphere with the radius R, then the transition is
approximately given by the following function (see Appendix D)
=1 if z 4 -R(1-a z2 R
a(z) = a = 1 - a) (1+-) (2--)/4 if -R < z < R , (2.11)
a =, if R z • z
This transition is an idealization, derived under the assumption that the
undulations Aa are zero within the particle aggregate, but it is not the
limit curve for R + o. (Such a limit is the constant a- (1+c)/2 for any
aggregate occupying a half-space). The real transition curve undulates
around a basis that is approximated by Eq. (2.11). The amplitude of the
undulations is bounded by Eq. (2.3) in which a is replaced by the basis
value. The approximation (2.11) is quite good even for moderate values of
R/Lm, as shown by the examples of transition curves in Section 4.
3. GAS VOLUME FRACTION AS A FUNCTION (F TOE AVERAGING VOLJUME
We consider the gas volume fraction a in an averaging sphere with the
radius R and center coordinate vector -. We assume that the particles are
spheres with radius s and arranged in a regular lattice with a lattice
constant L. Let a be the limit value of the gas volume fraction for an
infinite averaging volume. Then one can define (see Appendix B) a mean
distance Lm between particle centers by
Lm = 4s(1-I) -1/3 (3.1)
The gas volume fraction depends on the lattice type, the location X of the
averaging sphere with respect to the lattice, the radius R, the radius s, p
1"1
1,S
.........................
and on the lattice constant L. Because a is dimensionless, this dependencecan be expressed by a function of the following form
a = f(/Lm, R/Lm, s/Lm, L/Lm), (3.2)
whereby the function f1 depends on the lattice type. The ratio L/Lm is aconstant for a given lattice and, therefore, it can be included in thedefinition of the function fl" Also, because of the definition (3.1) theargument s/Lm can be replaced by a. Hence, one has the following twoequivalent representations of the functional dependence of a in a givenlattice:
a f 2 (1/L.1 R/Lm, s/Lk) (3.3)
and
x = f 3 (/Lm, R/Lm, a) (3.4)
Eqs. (3.3) and (3.4) show that a is a function of five dimensionless scalarparameters: the three components of the position vector I/L_, the averagingradius R/Lm and s/Lm or a. In this section we investigate the dependenceof a on the parameter R/Lm . We may think of this dependence as the resultof placing the averaging sphere at a fixed position I/L within a givenlattice and letting its radius R/Lm vary. If R is small, then one obtainseither a - 1 or a = 0, depending whether the fixed center position /LM isoutside or inside a particle. As R/Lm increases, a approaches the value
regardless of the center location. Figures 3 through 7 show typicalcurves of the function a(R/Lm) for four different lattices. (The latticesare defined in Appendix A and a was calculated with the algorithm describedin Appendix C).
Figure 3 illustrates the dependence of a on R/Lm in a square cylinderlattice with a - 0.5. The central graph shows two curves. One curve
corresponds to the center location coordinates I/Lm - (0,0,0) and startswith a=0, because the point of origin is a lattice point and occupied by asolid sphere. The other curve starts with a - I. It corresponds to the
center coordinates I/L - (0.4, 0.2, 0), which is a point occupied by gas.The upper graph in Figure 3 shows the same two curves in a larger scale.One sees that the undulations of a(R/Lm) around the limit value a have awave length of about one and an amplitude that seems to decreaseproportionally to a power of R/Lm . The lower graph in Figure 3 shows thenumber N of particles that are inside the averaging sphere also as a
14
0:
function of R/Lm . The two curves corresponding to the two locations of the
averaging sphere are almost identical and both approximate the cubic (see
also Eq. (B.6) in Appendix B)
N (2R/Lm)3 .(3.5)
Figures 4 through 7 present the same functions as Figure 3 but for
different values of a and for different lattices. One notices that the
qualitative as well as quantitative behavior of the functions are very
similar in all cases.
Our goal is to obtain an estimate for the amplitude of the undulations
of a about tho limit value a as a function of R/Lm . To that end one can
determine ror a fixed a and R/Lm the extreme values of a by treating
!L mas A free parameter vector. (One can envision this as moving them
avetdging sphere with a fixed radius in an infinite lattice until the two
positions are found that correspond to minimum and maximum a.) The
calculations were done with a simplex algorithm and some results are shown
in Figures 8 through 15. The figures show that there are differences in
detail between lattices and between cases with different a, but the general
trend of the undulation amplitude is rather uniform in all cases. This
trend is illustrated in Figures 16 and 17. Figure 16 shows in the left hand
graph the extreme negative deviations a-a from Figure 15 plotted over the
corresponding values of R/Lm in log,log-scale. The right hand graph shows a
similar plot of the extreme positive deviations. Both graphs of Figure 16
are combined into one graph in Figure 17. It is apparent from these figures
that the negative and positive deviations behave similarly and that their
amplitudes are in general proportional to (R/Lm) 2 . Plots of the extreme
deviations for other lattices look very much like Figures 16 and 17. A
combined plot of deviations in all four lattices for a = 0.5 is shown in
Figure 18 and for a = 0.9 in Figure 19. The different symbols in these
figures indicate different lattices and different signs of the extreme
deviations. The dispersion of the plotted points shows that none of the
considered lattices consistently produces larger undulations than other
lattices, and that neither negative nor positive deviations are consistently
larger within the R/Lm range considered. An estimated upper bound of the
deviations is shown as a straight line in the log,log-plots. It has the
equation
= 0.5 (I-)a 2(R/Lm) -2 (3.6)
Eq. (3.6) estimates a bound for the maximum amplitude of undulations for a
given size of averaging sphere. Using Eq. (3.5) one can also express the
estimate in terms of the number N of particles within the averaging sphere:
(Continued on page 33)
15
.--•
.."- . .- - " .- . . . . . . . .' % - -. - - 7 - . - - -, . - .- y
01
5654
E 52
-J-5
482
4
46 02 I I I I
0 123 4
I 30"4
- 10 2_
C')w
-0
<3.0(-U.
o 2.0
w1.0
0 .0
0.0
0 2 3 4R/LM
Figure 3. Gas Volume Fraction Dependence on Averaging Radius.
Square Cylinder Lattice, a - 0.5; s/1, 0.3969
16
a .- - .. ztv f. "... ct .. .. ..-
--
" 10 -2 I "1 1 1 I 1 1
94
93
92
0--J 91
90
89
88 I
0 1 2 3 4
010-I
8
< 6-
4
<4 j~i2
II
0
0 2 3
• 10 2
w, 4.0
Ot<3.0
o 2.0wIf
0.002 34
R/LM
Figure 4. Gas Volume Fraction Dependence on Averaging Radius.
Square Cylinder Lattice, a - 0.9; s/Lm - 0.2321
17
*
94.0 -93.0
92.0
9119. 0
90. 0
89.0
88.0 -87.0
86.0
0 1 2 3 4
*1 0 ' T
8
<- 6
< 4
2
0 1 2 34
.10 2 - --- - -jT j -T
LL I
.0
<3.0CL.
1 .0
0 12 34
R/L MrFigure 5. Gas Volume rrac:ion Dependence on Averaging Radius.
TraglrClne atca 09 /m 022
61
. . .. .* • ,' - o . -N
,. 4
[ . .010- 2":
96
9594
93= 92
< 91
90
89
88
870 1-2 3 4
8
< 6
-JJ.4 4
2
0
0 2 3 4
C 4.0
< 3.0 -
0-
~2.0F- .0
0.0
o 2 3 4R/LM
Figure 6. Gas Volume Fraction Dependence on Averaging Radius.
Leap-Frog Square Lattice, - = 0.9; s/L m - 0.2321
19
.0 . . .. : . _•.. .. , •. ... . . . .•. . . . . . .
01-
969594
A 93
-- 92
.~91
90
8988
* . 87
0 1 2 3 4
8 .10-1
8
* < 6
< 4
2
0
0 1 2 3 4
cn
< 3.0
02.0Fz z1.0
0 0.0
0 1 2 3 4R/LM
Figure 7. Gas Volume Fraction Dependence on Averaging Radius.
leap-Frog Triangular lattice, az 0.9; a/1a 0.232 1
20
53
<51
CL 49
47
45
43
41
0 1 2 34
8
< 6
< 4
2S
0
0 12 34
-12
(I)
C.(I.
C' 2.0
1 .00z
0 1 23
R/LM
Fi t.re_8. Extremes of Gas Volume Fraction as Functions of Averaging Radius.
Square Cylinder Lattice, a =0.5; s/L M 0.3969
21
-c -' 7-
56
< 52
< 50
__ 48
46
3 4
U .10-1
8
< 6
C-
< 4
2
0
o01 2 34
0102
LIU
~4.0
I:
-< ~3.01
c:)2.0
0:.0 1
O 2 34
R/Ltl
.4ire 9. Extremes of Gas Volume Fract ion as Functions of "veraging Racit
Triangular Cylinder Lattice, =0.5; s/L = 0.'196~9
22
r2r
62
60
58
56
a.< 52
50
I. 4846
0 12 34j
010-1
8
4< 6
<4
2
0
0 12 34
910 2
w
C-
Cr.0
0: .0
R/L1 .
,,,re 10. Extremes of Gas Volume Fraction as Fun, tions of Averaging Radius.
Leap-Frog square Lattice, a =0.5; s/L =0.3969
m
23
67
*10-262 -1I11II"
60
58
56
52
50
48
46 t"t44
0 1 2 3 4
8
* 6
-J
< 4
2
0
0 1 2 3 4]
J
._ 4 .0
* 3 .0 ---
CL02.0
*X
0.0
0 1 2 3 4
R/LM
f1igure 11. Extremes of Gas Volume Fraction as Functions of Averaging Radius.
Leap-Frog Triangular Lattice, a = 0.5; s/L = 0.3969m
24
.. . . ..- . . . . . .
*10 -2 _____________ __
93. 5
92.5
<90.5Q-
8 9-
5
<88. 5
87 .5
86.5
8.5 .5
84.5
0 1 2 3 4
8
S< 6
C-
< 4
6 2
0
0 1 2 34
- 10 2
-LJ
.0
< 3.0
1.0
LLL
o 2.0
-10-2
94.0
9:3.0
C 92.0~91 -0
CL
-90.0
89.0
88.0
86.0
0 12 3 4
< 6a-
<4
2
0
0 12 3 4
.10 2
L) 4.0
I-
< 3.0
U-
o 2.0
r1.0
0.0
0 1 2 3 4
R/LM
* ri-jr E 13. Extremes of Gas Volume Fraction as Functions of Averaging Radius.
Triangular Cylinder Lattice, a =0.9; s/L = 0.2321
26
95
93
CL. 91
89
87
85
610-1
8
* ,~ 6
"C 4
0 2 3 4
4.02
Ct,
U.
2-).0
0
0.0
0 12 3 4
R.'L M
Figure 14. Extremes of Gas Volume Fraction as Functions of Averaging R~adius.
Leap-Frog Square Lattice, a 0.9; s/L = 0.2321
27
07
95
-AL 91
89
87
85
0 1 234
.10- IJ
I I
L
8
6
< 4
2 i
0
0 1 234
.10 2
~4.0
< 3.0a-
Crw I
1 1.0
0.0
0 1 2 34
fjetire 15. Extremes of Gas Volume Fraction as Functions of Averaging Radius.
Leap-Frog Triangular Lattice, a =0.9z s/L = 0.2321
28
a'-*
T I I I I I f I
++ + +
+ +
+ +
++ +l
++
+ C;+ 0
SNOLIVIA30 VF/dIV ,-- 1ii 1iii i i 0 - ..C'.,q
u c
*- F- - -T I f I I 4 0
0 d
'-4
•-a "14
o 04
SNQIILA30VHCU
29)~ -
10--
0-
UiU
+
+t
.4
-J01
+/L
Fiur 7.Etrm Dvatos fGa olm Fatin
LepFo+ raglrLtie0.;sLm 022
U)
i o -2
CL
0---I 0
0 1 0R/LM
r
r Figure 18. Extreme Deviations of Gas Volume Fraction for All Lattices.
ai 0.5; s/L =0.3969
m
K 31
101I I I I !I i I
S
w 4
o• - 1 0 - 2
--
a- _ I
_.I X QFB4
L 4B
'10~
\S
, 00 10 1
R/LM
4 Figure 19. Extreme Deviation of Gas Volume Fraction for All Lattices
a= 0.9; s/L - 0.2321m
32
* I
(Continued from page 15)
A =2.0 _ ) 2 -2/3a 2.0 ~ N2 (3.7)
If one wants the undulations to be below a tolerance level then Eqs. (3.6)and (3.7) can he used to estimate the correoponding minimum size of the
averaging -;phere and minimum number of particles within the sphere,respectivel7. A discussion of such estimates and corresponding graphs arepresented ii Section 2.
Next we discuss the validity of Eq. (3.6) and (3.7). They are based on
sample calculations within the range 1.0 <R/Lm ( 4.0 (or 8 ( N < 512)
for i = 0.5, 0.667 and 0.9, and tor four different lattices. Because thefour lattices have quite different symmetries and maximum packing densities,ore can a sume that the results are representative for all reasonably
regular an,' uniform particle arrangements. One can of course constructperiodic particle arrangements for which the undulations are in excess of
the estimated limit at some point. However, in order to do so, one would
have to stretch the concepts "unifoum" and "regular.- The calculations
cover all a values of practical importance, that is, all a values over0.5. (The formulas (3.6) and (3.7) both have the correct limit
= 0 for a = 1.) We notice that the minimum value of a (closet packing
of spheres) is between 0.26 and 0.48 for the four lattices considered. (See
Appendix A.) Therefore, values of a below 0.5 are only possible forspecial arrangements of tue particles, and extrapolation of the general
formulas to a below 0.5, should be done with reservations. The range ofthe parameter R/Lm between 1.0 and 4.0 was also chosen by practical
considerations. For values of R/Lm below one the undulations are large and
Srregular, indicating that the corresponding average is not a goodrepresentation of a particle aggregate. On the other hand, at R/Lm = 4 the
undulations have an amplitude of the order 10 - 3 , which is sufficient formost applications. Extrapolations of the formulas to larger values of R/Lm 0
likely will give correct estimates, but they have not been tested by sample
calculations.
.4., "
4. GAS VOLUM FRACTION PROFILES
In this section we give an example of gas volume fraction profiles,
i.e., of the dependence of the gas volume fraction a on the trajectory ofthe averaging sphere, and of the smoothing effect of averaging on a flow
structure. The example is a particle aggregate with a uniform limit gas
volume fraction a that occupies the half-space z ) -s. (By a uniform limitgas volume fraction we mean that the particles are arranged in a regularpattern (lattice) with the limit value a for that part of any infinite
averaging sphere that is inside the aggregate.) Let the particles be
spheres with radius s. We define the boundary of the aggregate as the plane
z = -s, that is, the tangential plane to the border particles with a zero z-
coordinate of their centers. A naive representation of the particle
aggregate in terms of a is by the step function
CL I if Z -S (4 1= _ (4.1)
S if z > -S
However, this representation is not correct because the gas volume fraction
is only defined in terms of an averaging volume. (The gas volume is by
definition an a-fraction of an averaging volme.) The step function (4.1)
does not correspond to any finite or infinite averaging volume and,therefore, cannot be a representation of the aggregate boundary. A true
representation of the aggregate has to include a size parameter of theaveraging volume, in our case the averaging radius R, and one obtains
different transition curves between a = I (outside the aggregate) anda f a (inside the aggregate) depending on R, on the trajectory of theaveraging sphere crossing the boundary, and on the arrangement (lattice) of
the particles.
Figure 20 shows examples of transition curves for two differentaveraging spheres and two trajectories for each sphere. The spheres havethe radiuses R/Lm = 1.0 and R/Lm = 2.0, respectively, and the two
trajectories are along the z-axis and along the line x/Lm - 0.5, y/Lm =
0.5. The particles are spheres with radius s and they are arranged in aleap-frog triangular lattice. (Lattices are defined in Appendix A.) Figure
20 shows the transition curves 'in two different scales. The general slopes
of the transitions are about (1 - a)/(2R/Lm), but their details depend on
the trajectories. Inside the particle aggregate one observes periodicundulations with an approximate wavelength of 1.5 Lm about the limit value
a . The amplitude of the undulations is bounded by Eq. (3 .6) and it of
course does not decrease as the averaging sphere is moved deeper into theaggregate.
34
-
-i0-1 10 1I I I I I I I I I I |
9.5 iS
8.5
a- 7.5
6.5
5.5
4.5
-3 -2 -1 0 1 2Z/LM
.10-1
8 -
6-"
0 l I l I I I I I I I
-3 -2 -1 0 1 2 3. ZILM
Figure 20. Gas Volume Fraction at a Particle Aggregate Boundary.
= 0.5; s/L m = 0.3969R/Lm
= 1.0 and 2.0Leap-Frog Triangular Lattice
35
S
7 .. . * -
2
Figures 21 and 22 show transitions along the same trajectories and thesame lattice, but for other particle sizes, that is, for other limit valuesa . The genural patterns of the transitions and undulations are the same asin Figure 20, and they are typical for all four lattices considered in thisreport. The wavelength of the undulations was found to vary between 1 .0 I m
and 1.5 Lm, depending on the particle arrangement.
An approximate formula for the transition curve can be derived underthe assumption that the undulations within any subsection of the cloud are p
zero. Then the curve is (see Appendix D)
.1 if z + s < -R ,1
) -0.25 (1-a) [l+(z+s)/R 2[2-(z+s)/R] if -R < z + s < R L4.2)
a if R< z+s
X reprts,,ntation of the particle agg. gate by the function (4.2) is betterthan Fq. ( .1), because the former !:akes into account the averaging
radius. However, since undulations have been neglected, any real transitioncurve will deviate from (z) by undulations that are bounded by Eq. (3.6)where A is to be replaced by (z) . In addition to the undulations, other
deviations can be present due to a general structure of the particle
arrangement
The transition curve a(z) with bounds computed by Eq. (3.6) andcorresponding extreme values of a are shown in Figures 23-25 for a leap-frogtriangular lattice. The extreme values were computed for fixed z-values bya simplex algorithm for which the coordinates x and y were treated as free
parameters. Conceptually this means moving the averaging sphere within theplane z=const. until the two positions with extreme values of a have been
found .
According to Figure 23 the approximation a(z) is quite good in thisparticular case. However, if the particle size is reduced, as In Figure 24,
then a systematic deviation becomes apparent in the upper part of thecutrve. This deviation is also visible in Figure 25, that is, it does notdisappear with increased averaging radius. Such deviations were alsoobtained for other lattices, and they can be positive as well as negative,depending on the lattice and on the orientation of the boundary.
5. DISCJSSION OF THE RESULTS
The sample calculations in Section 3 and 4 show that the gas volume Jfration a Is a function that undulates about a limit value a with a
w-veiength 1 .5" L (Lm is a mean distance between particle
(Continued on page 421
36
" , / "i' °
9.0
T 8.6
7.8
7.4
7.0
6.6
'-3 -2 -10 1 2 3Z/LM
010 -1
6
2
-3-2 -10 123Z/LM
LiLLIi Ga s Volomi.- Fracrton ,t a Particle Aggregate Boundary.
0 .7; SI/' = 0 .3347
R/Lm= .0 and 2.0
Leap-Frog Triangular Latt ice
17
*10-2 1 1 1
99
97
95
-93
91
89
87-3 -2 -~0 1 2 3
8
6a-
4 4
2
0
-3 -2 -1 0 1 2 3Z/LM
Fi,,itrt, 22. Gas Volume Fraction at a Particle Aggregate Bounda ry.
I=0.9; s/Lm = 0.2321
R/Lm = 1.0 and 2.0leap-Frog Triangular lattice
30
0.
7.0\
600
S <7.0
4 L --- 2 ----
-3-2 -10 1 2 3
Z/LM
6X
4 F- r
I. L L iK L L L L. L
-5-2 -I0 1 2 3
Z/L Mi
*~ ~ I -. it... y-,,i Bww2~'ids of Gas Vowrne Fracti-fl
1,, Ag.g~. V -) B .ndary
I" lil~ !v - r t
010-2
99
97 0.
95
CL 93-J
91
89
87F
85 _____________________________________
-3 -2 -10 1 2 3Z/LM
010 I I1-1ijj
8
6
0I-
4 -
2
-3 -2 -10 1 2 3Z/LM
4 FiguriEL24. Extreme Values anid Estimated Bounds of (,.is Volume Fract ionat a Particle Aggregate Boundary
a = 0.9; s/I,. = 0.2321RIL = 1.0m
mLeap -Frog Triangular Lattice I
40
.10-2
98
96
94
940
-3 -2 -1 0 1 2 3Z/LM
010-1 I*
8-
6
4 4S
2k
-:3 Z/LMj
25 Extreme Values and Estimated Boundq of Gas Volume Fractionia: a Particle Aggregate Boundary
0.9; S/Lm = 0.2321
R / I' = 2 .0IIfeap-Frog Triangular Lattice
41
(Continued from page 36)
centers) and with an amplitude bounded by Eq. (3.6). Now we discuss
consequences of this behavior of a for the representation and computation of
a flow field. First, we notice that in an averaged flow field any flowstructure with extensions less than R (the radius of the averaging sphere)
is smoothed out and also possibly masked by the undulations. Second, the
minimum size of R depends on the demands one makes on the averages.
Obviously one can only consider an average as a property of a particleaggregate if there are many particles representing the particulate phasewithin the averaging volume. This remark is quantitated by the estimate of
undulation bounds, Eq. (3.6). As the discussion in Section 2 shows, one
needs 30-150 particles to keep undulations of a below 0.01 and at least 30
particles if the limit is 0.03. The corresponding minimum values of theradius R are 2.4.L M and 1.8. Lm, respectively. For the sake of the following
discussion we assume that R is chosen to be equal to or larger than 2. Lm .
The findings have the follow ig consequences for the representation ofa two-phase flow field by average fh 1. parameters. Because one cannotdistinguish between noise (particle induced undulations) and structures
4 which have extensions less than R, one has to classify as noise any flowstructures that extend less than Lm . Consequently, if a flow field is
specified in a mesh, the mesh constant need not be finer than Lm . Any mesh
refinement can be done by interpolation without loosing accuracy. The sameapplies to experimental data. If measurements are done locally with a finemesh then the results should be averaged over a volume corresponding to R =2-Lm at least. This applies, of course, also to measurements nearboundaries where the averages again represent a finite averaging volume, andnot a local point.
Similar consequences can be drawn for the computation of two-phase flowfields, for instance, by solving average flow equations. A reasonable |
computing mesh constant is of the order of Lm . Refinement of the meshconstant below Lm has at best the effect of interpolation and, at worst, itmay induce noise into the solution. in any case such a refinement wouldwaste computing time. If flow structures with extensions less than R appearin the solution, for instance, in mesh refinement studies, then they shouldbe considered as numerical artifacts, because they cannot be interpreted as
average flow properties.
These considerations have an interesting result for transient flow
fields in which a + I in some parts of the field. If a approaches onebecause the particle sizes are duced (for instance, by combustion) then
the condition R > 2. Lm is not violated and the particulate phase can berepresented by averages. If, however, a approaches one because particlesdiffuse from the mixed phase region into gas-only regions, then the
condition R > 2. Lm will be eventually violated and the average model
4 invalidated. A correct approach in such a situation is to model the
rarified parts of the particle aggregate by some other method thanaveraging, for instance, by individually tracing the diffused particles.
42 ..
In Appendix B we discuss the value of the mean distance L. betweenparticle centers and conclude that in typical interior ballistics problems
Lm caai be as large as three initial propellant particle diameters.According to Figure 1 one then needs an averaging sphere radius that isequal to 6 to 8 initial particle diameters in order to keep the undulationsbelow 0.01 . For some interior ballistics situations this means that no flow
details in radial direction can be represented by a two-phase theory that isbased on averaging, because the minimum diameter of the averaging sphereapproaches the diameter of the barrel. In such situations the two-phasetheory only can model averages taken over a cross-sectional segment of the
barrel, that is, the theory can be used to compute the interior ballistics
core fLow.
43
APPENDIX A.
IATTICESS
45
I:I
APPENDIX A
IATTICKS
For the calculation of the gas volume fraction o one has to specify the
sLze and the spacial arrangement of the particles. In this report, the
arrangement is assumed to be in the form of a three-dimensional lattice. In p
order to assess the significance of the arrangement, four different lattices
were used for the calculations. These lattices are defined in this
appendix.
The square cylinder lattice is shown in Figure A.1 . It may be
constructed by arranging the particles in a square pattern with a mesh
constant L in the x,y-plane, and translating the arrangement in the z-
direction by multiples of L. Each original square thereby generates a
square cylinder. The lattice is identical to the well-known cubic lattice.
4 We identify each lattice -oint by three integer parameters, ij, and
k. The Cartesian coordinates of a lattice point with the identification
(i ,,k) are
x(i,j,k) iL I
y(I,j,k) = jL , (A.1)
z(i,j,k) = kL
*DThe centers of the particles are assumed to be located at the lattice
pont.1 Let :- be the radius of a particle. If the square cylinder lattice
icoupLes ar infinite space, then the corresponding gas volume fraction is
- -4 ( A3~j7T ~ (A.2)
r -,alle,;t value of a is obtained if the particles are as large as
4i l- Ie. Because s cannot be larger than L/2 without intersection betweenprtilies, one obtains from Eq. (A.2)
rx = 0.476 401 2 (A.3)
4p7
47
I
I+I
I
L5
Figure A.1. Square Cylinder lIattice
YS
LI
FiueA2 Triangular Cylinder latctce
48
• ' " "
The triangular cylinder lattice is shown in Figure A.2. It can be
constructed by arranging the particles in an equilateral triangular patternwith the mesh constant L in the x,y-plane and translating the arragement by
multiptes of L in the z-direction. Each triangle thereby generates atriangular cylinder. Again using the integer triple (ij,k) for theidentification of each lattice point, one obtains the following formulas forthe Cartesian coordinates of the points:
1
x(ij,k) = iL + Ij (mod 2)1 - L
y(i,j,k) j V L (A.4)2
z(i,j,k) = k L
in a triangular cylinder lattice that occupies an infinite space the gasvolume fraction is
a=1 8 sL 3= 1 - 8 (A.5)
where s is the radius of tie particles. The smallest possible value of u is:abtalned for the largest possible value of s. Because s cannot be largerthan L/2, one obtains
m I.= I -/(3/) = 0.395 400 2 (A.6)
The leap-frog square lattice is shown in Figure A.3. It can beconstructed by first arranging the particles in the x,y-plane in a squaremesh with the mesh constant L and the square sides parallel to thecoordinato axes. Then the pattern is translated by multiples of L/vo2 in thez-direction and by multiples of L/2 in the x- and y-directions. Thus, eachspiare is translated in a leap-frog manner from one z-plane to the next.The ensuing lattice is also known as "face-centered cubic lattice" with thel.ittice constant 2. L. Identifying each lattice point by the integertriple (i,i,k). one obtains the following expressions for the Cartesian
coordinates of the points:
( ,j ,k) = i L + 1k (mod 2)1 1 L
y(ij,k) j L + Ik (mod 2)1 1 L (A.7)
7(i,jk) = k - L
49
I"
L r2
L LiFigure A.3. Leap-Frog Square lattize
L66
lepFo6ranua atc
50
:
If a leap-frog square lattice occupies an infinite space, then the gas
volume fraction is
= . (A.8)
Because the maximum possible value of the particle radius s is L/2, one
obtains from Equation (A.8)
a = - 1T/(3 /f) 0.259 519 5 (A.9)
The leap-frog square lattice and the leap-frog triangular lattice, which wedescribe next, have the highest possible particle concentrations (the lowest
possible a min ) of all three dimensional lattices.
The leap-frog triangular lattice is shown in Figure A.4. It can beconqtrurted by first arranging the particles in the x,y-plane in anequilateral triangular mesh with the mesh constant L, as shown in Figure
\.2. Then the pattern is translated in the z-direction by multiples of /2-3 L,and in the y-direction alternatively by ±(/3/3)L. Thus, the triangular meshis shifted in a leap-frog manner as one proceeds from one z-plane to theievt. Identifying each lattice point with the integer triple (ij,k), one
)btains the following Cartesian coordinates of the lattice points:
X(i,j,k) = i L +j (mod 2)1 . L
y(i,j,k) = j -- L + fk(mod 2) ---L (A .10)
2z(i,j,k) = k / L3
T>,.'s vol-ime fraction a in an infinite leap-frog triangular lattice can be
computed by the same formula (A.8) as for the infinite leap-frog squarelutti-e. Consequently, also the smallest possible amin is given by Eq.(A .9 ).
The foranjas for a have for all four lattices the form
03- I - A (s/ L) , (A. I)
where A is a constant that depends on the lattice type. Because we havedefind the lattice constant L in such a manner that it is the closestdist-rice berween any two centers of particles, the minimum value of a is inill cae,; obtained by setting s = L/2 in Eq. (A.11). Iherefore,
09
F '. s -' *-
A 8 (1- a ) =8 a (A. 12)
where is the highest possible solid volume fraction in an infinitemax
lattice. Eliminating A between Eq. (A.1i) and (A.12) one obtains
a 1 - 8 (s/L)3 (A. 13)max
as a general formula for a. The value of a depends, of course, on themax-
lattice type. Table A.I lists the values of amin and 8 for the fourmax
lattice types considered. The last column of the table contains the ratio
of the mean distance between particle centers and the neighbor distance.That ratio iq also a character4 stic of each lattice. The mean distance is
defined in Appendix B.
2
*
S
152 '.S
QXI)
-r-.
~ 000 (L)
F- ~0 00
'A 00- - -
-4 00 -I 0
43 CDI -T -T
zo .o 0f if
0 0 0C>'
Co)
*nV(N-IC'
tv totk4w0 0 1-4
1.4 -4 11o a" -
0 3 1 a-. Et
w5
0 .
55
I
p
I p
p
I p
a
I
I I
I I
I I
-j
APPENDIX B
NEIGHBOR DISTANCE AND MEAN DISTANCE
The density of a particle aggregate can be characterized by various
parameters: the solid volume fraction B, the particle volume and thenumber density, the particle radius and a mean distance, the particle radius
and a neighbor distance in a given lattice, etc. In this Appendix we
discuss the concepts neighbor distance and mean distance.
As neighbor distance we define the smallest distance between particle
centers. For the lattices defined in Appendix A the neighbor distance isequal to the lattice constant L. However, the number of neighbors with adistance L is different for different lattices. Thus, in a square cylinder
lattice each particle has six nearest neighbors, in a triangular cylinder
lattice the number is eight, and in the leap-frog lattices there are twelvenearest neighbors to each particle. One can also construct regular latticesin which the number of nearest neighbor., is not the same for all particles.
For any of the four lattices considered in this report one can
calculate for given a the corresponding neighbor distance by solving Eq.
(A.13) for L:
2s (8max 2s( max B.1)
M1Te values of 6max are given in Table A.1 for each lattice. We see from Eq.max.
(B.1) that the neighbor distance is different for different lattices, evenif s and a are the same. This is not very convenient when computations arecompared. Therefore, we also define a mean dista.ce Lm that is independentof the arrangement of the particles.
Let the nuimber of particles in the particle aggregate be m, the volume3of each particle be v(s) = (4/3) 7T s , and the volume of the particle
aggregate be W. One may conceptually assign to each particle the fractionW/m of the totil volume and represent this fraction as a virtual sphere withthe diameter n" This diameter we define as the mean distance between the
,-enters of particles. A relation between the solid volume fraction B and L.
carn be derived as follows. By definition one has
W T L L3 (B.2)
and the solid volume fraction in W is
57
m V(s) _V(s) L3= w v- 8 (s/L , (B .3)
or
L s/ 2s(I-)-/ . (B .4)m
In Eqs. (B.3) and (B.4) we have used 3 and a (the limit values) because the
definition (B.2) pertains to the whole particle aggregate. We notice that,' -."the quotient W/m is the inverse of the number density of particles.
* Therefore, Lm is proportional to the cube root of the inverse of the number
density. If we consider a finite part of the particle aggregate, defined byan averaging volume with radius R and containing N particles, then the
equation corresponding to Eq. (B,:) is only approximately valid:
(2R) 3 I
- L (B.5)N 6 m
(The equation is not exactly satisfied because LM is the mean distance forthe whole particle aggregate.) Solving Eq. (B.5) for N one obtains
N ' &-3- (B.6)m
as an estimate of the number of particles in an averaging sphere. Theapproximation by Eq. (B.6) is quite accurate, as shown by the numerical
results in Section 3. The relation (B.6) is of course independent of the
particle arrangement, i.e., of the lattice type.
The advantage of the mean distance LM over the neighbor distance L isthat the former is independent of the spacial arrangement (lattice) of theparticles. It is a parameter, like a, a, s and the number density, thatdefines a general characteristic of the particle aggregate. Because 8 = 1-a
and because Eqs. (B.2) and (B.3) hold between these five parameters, only
two of them can be prescribed for a particular aggregate. In addition to
the two parameters, one may also specify a structure (lattice) of the
partici- arrangement, which includes a definition of a lattice constant L.The relation between the lattice constant L and the mean distance Lm isdifferent for different lattices. For the lattices defined in Appendix Aone obtains with smax = L/2 from Eqs. (B.1) and (B.3)
L/ 1m 1/ 3max (B.7)
58
"0 : - - ': , - - : . .- - " ,, " . . .. . .• . . " . .
--o -- | . I ~% ! % q m p p C
*
The value of max depends on the lattice type. Numerical values of the' ratir Lm/L are given in Table A.1, and according to that table the mean
distance in the four regular three dimensional lattices is 10% to 24% largerthan the neighbor distance.
Next, we estimate the mean particle distance for interior ballisticsapplications. Obviously any average distance for the whole gas-particle
system increases during the firing cycle, because the available volumeincreases due to the motion of the projectile. Let a 0 be the initial gas
0Ivolume fraction in the gun chamber and so be the initial radius of the
particles. Then the initial mean distance is according to Eq. (B.4)
Lm = 2so(i t )/3 (B.8)
Let zchamber be the length of the chamber and Ztravel be the travel of theprojectile from its initial position. The ratio of the available volume W.before and W after the projectile has moved by Ztravel is approximatelyI + ztravel/Z lamber. The corresponding relative increase of the meandistance Lm is according to Eq. (B.2) the cube root of this ratio.Therefore, we have the following general formula for the mean particledistance in a gun:
Lm = Lmo(W/Wo)1! 3
1/3 -1/3= 2s (t+Z /Z) (1-ax (B.9)a travel chamber o
According to this equation the maximum of the mean distance is reachedduring a firing cycle when Ztravel is the muzzle value. Typical numericalvalues in Eq. (B.9) are as follows. The initial gas volume fraction ao isbetween 0 .4 and 0.6, and Ztravel/Zchamber ) 10. Therefore, the maximum ofthe ratio Lm/2so is for a typical gun between 2.64 and 3.02. Differently1Flrmuatied, the maximum of the mean distance Lm between particle centers isfor interior ballistics typically about three initial particle diameters.
59
II
-159I
6I| . . " m m
I
S
S
S
S
S
5
I
S
I
APPENDIX C
AILORIlIM FOR GAS VOLIEM FRAM~ON CALCUIATI(!N
a6
-' - d.,.-.......~'.1. - *~- . - .' -. ' -, . -
V. .
I
iiiI
I I
I
I
I I
* A4
- . j -- -; .. -- . -
APPENDIX C
AIWORIfM FOR GAS VOLIUM FRACTION CALCUIATION
The gas volume in the averaging sphere with radius R can be calculated
by first computing the sum of the volumes of all particles that are located
inside the sphere and subtracting the sum from the sphere's volume. For
particles that are intersected by the surface of the averaging sphere one
only has to add to the sum the volume of the intersection between the two
spheres. Next we compute the intersection volume. According to Figure C. 1
we have the following relations if two spheres intersect:
A +a =D
A2 + h 2 R2 (C. I)
9 2 2+ h2 s
Eliminating h2 and solving for A and a one obtains from Eq. (6.1)
Center of o
Averaging Center ofSphere Particle
Figure C.I. Intersection of Spheres
2 R 2(c.2)
l 2 2 2-(
2 ITa* 2,-
*6
low"I
The volume of the intersection is the sum of two spherical segment
volumes. The segment of the averaging sphere has the volume
VR R-A) [3h2 + (R-A) 2 ] (R-A) 2 (2R + A) , (C.3)
the volune of the particle segment is
Vi (s a) 2 (2s + a) (C.4)
and the volume of the intersection is
Vi = v R + Vs . (C.5)
The computation of the solid volume within the averaging sphere can be
done as follows. First, using the given coordinates of the center of the
sphere Lnd the specifications of the lattice, one defines a search area in
terms of: minimum and maximum values of the integers (i,j,k) (see Appendix
A). Next, one computes for each lattice point within the search area the
distance D between the point and the center of the averaging sphere. If
D > R + s, then the particle is outside the averaging sphere. If D 4'1R -
then on,! of the two spheres (averaging sphere and particle) is inside the
other, dpending on which one is bigger. Then the solid volume is equal to
the sma'ler of the two sphere volumes. Finally, if IR-sl<D<R+s, then the
interse, tion volume is calculated by the formulas (C.2) through (C.5) and
added t) the sum of solid volumes that are located inside the averaging
sphere.
6I
64
_.
APPENDIX D
GAS VOLUME FRACTION AT A PARTICLE AGGREGATE BOUNDARY
0
65
S1
- -. --- .- -* * ~ "~- -~ n - ** -'. -
4 I
I
I
I
a I
*
I ,1I
I I
APPENDIX D
GAS VOLUME FRACTION AT A PARTICLE AGREGATE BOUNDARYGILet a particle aggregate with the limit gas volume fraction a occupy
the half-space z > 0. Our goal is to obtain an approximate transition of
the gas volume fraction from a = I at large negative values of z to a = at
large positive values of z. In order to derive the approximation we assume
that the gas volume fraction can be approximated by at in that part of the
averaging sphere that intersects with the half-space z 0 0. Under this
'assumption U ii only a function of the coordinate zc of center of the
* averagi ng spher O. ur goal is to find that function.
We notice that the assumption about the approximation of a by a cannot
4 be realized in any real particle aggregate. The undulations of a about
Can be reduced to arbitrarily low levels if R/L. is sufficiently large, but
ul[y if the whole averaging sphere is within the half-space z ) R. If the
averaiing sphere is allowed to intersect partly with the half-space z 0,then the intersecting volume cannot be assumed always large, even if R/Lm >
1, and, theref )re,the undulations are only bounded by IAI4 max(a, I-a).
Let z bi, the z-coordinate of the center of the averaging sphere. Thevolume of the intersection of the sphere with the half-space z > 0 is
0 , if z -R
V (R + z z) , if -R < z < R (D. I)3_ R zc c c
4 R3
R , if R 4 z
Because we assume that cx ax in any finite part of the half-space z > 0, the
gas volume fraction in the averaging sphere is
i V- + (V R - V) 1I,
VR (D.2)
i - I - ) V-/VcxR
wt', yR (4/3) -i R3 is the volume of the averaging sphere. Combining Eqs.
I1) and (D.2) one obtains
67
I I
I if z 4 -R2 C
a(z ) = 1-(-a) (1 + z /R) (2 - z /R)/ if -R < z < -R (D.3)C C C C
a if R< zC
The limit of the transition curve for R/Lm+ * is simply aZ-(I+)/2.
This is a nice illustration to the smoothing effect of averaging: If the
averaging volume is made so large that the particle-induced heterogeneities
are zero, then also all flow structures (aggregate boundary in the presentexample) are reduced to constants.
68
I
0
S
0]
. . . . .* . .
I I
I
I I
I I
II
I
a
II
I I
II
II
APPENDIX E
UNDULATIONS OF AVERAGE FUNCTIONS
Let ¢ be a local gas property, for instance, density. Then the
corresponding average ., is defined by
= r p dV ,(E.1)
Vgas 'V
gas
where V is that pai-t of the averaging volume V which is occupied by
gas. I is constant, then obviously t zt for any positive value of Vgas
aV. If $ is not constant, then the particles in V influence the value of
the average *, that is, depends in such a case on a. Particularly, if
,% undulates, then also ) undulates. We estimate the amplitude At of the
undulations for a local function , with a constant gradient. We assume
without loss of generality that only the x-component of the gradient is non-
zero and express € by
4(K,y,z) =o + (X-xo) x (E.2)
with constant xo and . The corresponding average value is obtained by
substituting the expression (E.2) into the integral (E.1). Let the particle
volumes v be all equal and small compared to the averaging volume V, let x
denote the x-coordinate of the center of V, and xi, i=l,...,N, denote the
c:enters of the N particles that are inside the averaging volume V. Then the
iverage value of I is approximately equal to
NNp(x) wc ~ +(x-x v N X
jv rV-Nv (x )X - v 4 (xi-x)} (E.3)
+ (x-x ) " - ( /ao o x x
* whire i,; the, average deviation of the particle positions from the center,of V.
Next we relocate the averaging volume to the position x+Ax. Therelocation affects the average value of $ in two ways. First, the local
function is averaged over a different region and the positions of the N
particles have changod relative .' V. According to Eq. (E.3) this changes
71
the average by
6CAx* x(I + (1-a)/c*) (E.4)
A second effect is due to a possible change of the number of particleswithin V. let AM be the number of particles that have been added by therelocation and Am be the number of particles that have been lost by therelocation. The total number of particles within V then chaL.7es by
L N N AM - ',m (E.5)
C. ~ And the gas volurre fraction chaL-Iges by
= AN =-(1-a)LN/N (E.6)V
The average value of at x+A'x is, with the same approximation as Eq. (E.3),
4(x+AX) =V-(N+AN)vp JV'Li + (x+Ax-x )4] -
N AM Am-v ~ O ~ ~- (xi -x x
V-NsANv [V-(N+AN)v] + (x+Ax-x )~-(E.7)
N AM Am
- I( + - [xL~- X- Ax,
U = ~+ (x+Ax-x -
AM AmnL+a x[EAx +~X - ) (xi -x- Ax)]
We consider the last term in Eq. (E.7) and observe that the particles areadded and removed at opposite sides of the averaging volume. If V is asphere with radius R,then one may estimate the average x-coordinates of theadded particles by xif x + 2R/3 + Ax/2 -and the corresponding average of thecoordinates of the lost particles by xic x -2R/3 + Ax/2 . Then the lastterm in Eq. (E.7) is
72
l -4
1 AM 1 Am
N (x1-x-Ax) - - (xi-x-Ax)
_ INAM(2R/3 - Ax/2) +- Am(2R/3 + Ax/2) -- (E.8)
= --R(A M + AM) - Ax (AM - AM)N 3 N2
Next we express AM and Am in terms of a and Aa. The region of space newly
covered by the relocation of V has the volume
AV =TRAx = V 3Ax (E.9)4R
and an equal volume is abandoned. Let aM be the average value of a in the
added region and o be tle average value in the abandoned region. Then the
numbers of added and lost particles are
l-L
AV V 3Ax M 3& xAM = (1-a ) -V (1_uM) V x M - N -x
A v M v 4 - 4R
a nd (E.10)
1-a 1-aAV _ i m 3A xIn (-a) - - N
m v v 1-a 4R
Substituting these values into Eq. (E.8), one obtains the expression
I-a +1 3a-a a -amM Ax + (E.)
2( ) 8R L-a
It the g-is volume fraction is uniform (aM a s a), then this expression is
,qual to A x. We assume that a is not uniform, particularly that a is
iicreased by Aa due to an excess of added particles over the average AM in
Fq. (E.10fl. We express this assumption by modifying Eq. (E.1O) to
AV 3AM= (I-,) -+AN N - Ax +AN
v 4R
and (E.12)
Am = (I-Q) = N 4 Axv 4R
73
,0 . . - .. . . - _ , , i . .. ., -, i_ . _..
Then the expression Eq. E.8) becomes equal to
A x + AN A-Ax AN (
Replacing the last term in Eq. (E.7) by this expression we obtain
(x+AX) , + (x+A X--X )ix -
(E.14)
-- 1-a AN 2R Ax
or, using Eq. (E.6)
p(x4Ax) =0 + (x +Ax-X o ) X
(E.15)
_-, Aa 2R 3 Ax
a+Ac x 1-3 22R
The difference between Eqs. (E.3) and (E.15) is
S(x-4-Ax) W A(x X =
(E.16)
I 2R 3 A--a 1 ---+'--3a 2R a 2R
Hence the change of the average function e due to a change of a is
Aa --x + 3 -] (E.17)C-20 x 3a-+a 2I- 2R --- R]
In order to estimate the amplitude of the undulations of we set Ax equal
to about one fourth of the wave length of the undulations of a. According
to Section 4 the wave length is between 1.0 and 1.5 Lm . By setting Ax -
Lm/3 in Eq. (E.17) we obtain
747l, ::::S
-°' - .* --. 4 .. :: .. :1.;ir -, , _ : ... . . - . .; . , .. .- - -- •- f•-.
LC -2 R An I + a 3(E.18)x 3(a-Mn) L 4R a 2R] E.8
The quantity 12Rx is the maximum change of the local gas property * along
a diagonal of the averaging volume. According to Eq. (E.16) this is also
the change of the average function 0 if undulations are not present. Let
that change be denoted by 6 . The term Lm/4R in Eq. (E. 18) can be assumed
less than 0.25 or smaller. Also, J] <- 2R, so that the two terms involving
Lm and can be neglectel in our estimate of C. If we also assume
I Aal << a, then Eq. (E. 18) yields the following estimate for the amplitude Apof the undulations of 0:
-II/(3a) .(E. 19)
A corresponding estimate with 1-a replacing a can be obtained for the
average properties of particles, which are defined by
I $dV (E.20)
partc. Vpartc.
where Vpartc is that part of the averaging volume which is occupied byparticles and p is a local particle property.
Undulations of a also can affect such average functions which have azero gradient ( ¢=O ), if the local property is affected by the presence
of particles. One example of such a situation is a particle aggregate that
noves with velocity u through a quiescent viscous gas. Then the average gas
velocity in a large averaging volume is not zero but approximately a,fraction of u. The size of the fraction depends on the thickness of the
boundary layer around each particle and on the number of particles, that is,
on a. A similar example is the average temperature of a gas in which one
places particles with a different temperature. We notice that in the latter
example the average gas temperature undulates with an amplitude proportionalto the undulations of a, whereas the average particle temperature isaffected only by higher order terms. We demonstrate this remark byconsidering the following simple model in which the gas has a constant
property 'tb in a boundary region around each particle and a constantproperty o elsewhere in the flow field. The volume of the boundary region
around a particle we denote by v. (For a spherical particle with radius sand boundary region thickness As << s one has e 3As/s.) Then the average
gas property p is
4p
75
.-1
-. 4
S[(V-Nv -Nv c) + Nv eb]
+- (1--) o (1-ct EbI (E.21)
-o + (b -o (l-a)/c
Because the average depends on a, undulations of a will cause undulations
of . For small undulations Aa << a one obtains from Eq. (E.21) the
estimate
A b = 2) Aal/ 2 (E.22)
that is, Ap is proportional to Aa. The average particle properties are not
affected by Aa under similar conditions. Let b be a constant particle
property in a boundary region within each particle, and 0 be a constant
particle property elsewhere in the particles. The average ip then is
= -v[I Nv - Nv c ) + Nv b
1- [(-a - (1-a)) o + (1-a);b] (E.23)
- + b )c0 b 0
Because the average p is independent of a, it is not affected by undulations
of a.
To summarize, we observe that if particle induced undulations of
average flow properties do occur, then they likely are proportional to the
undulations a of the gas volume fraction.
76
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Director
US Army Ballistic Research LaboratoryATTN: AMXBR-OD-STAberdeen Proving Ground, MD 21005-9989
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