AN ABSTRACT OF THE THESIS OF
Varna Lindqvist for the degree of Master of Science in Mechanical Engineering
presented on September 13, 2011.
Title: Simulation of Drag Reduction due to Multi-Aligned Spheres.
Abstract approved:
____________________________________________________________________
Deborah V. Pence
James A. Liburdy
A method was sought to predict the flight paths and collisions for closely
spaced ink droplets of various sizes as a design aid for ink-jet printing development.
Computational fluid dynamics models of two rigid aligned spheres, as a proxy for ink
droplets, were initiated in atmospheric pressure air at constant velocities of 6 m/s to
14 m/s, and drag coefficients were calculated from the forces which developed on the
spheres. Simulations were run for the Reynolds number range of 5 to 25 for equal
sized 20 μm and 26 μm sphere pairs at center-to-center separation distances of 1.5 to
19 sphere diameters. For separations of 5 diameters and less, where relative sphere
size was found to influence the drag, simulations were also conducted using smaller
trailing spheres of 0.6 and 0.8 times the diameter of the leading sphere. Empirical
equations were found for the drag coefficients as a function of separation distance,
which also incorporated a factor to account for unequal sized spheres. From these
equations a calculator was developed to estimate the sphere trajectories, and the
collision time and distance, for two spheres given the initial diameters, velocity and
separation distance. The calculator predicts that in a 2 mm distance between an inkjet
cartridge and the paper, equal sized spheres will collide when separated by no more
than 3 diameters for the 14 m/s, 26 μm diameter case and up to 5 diameters for the 6
m/s, 20 μm diameter case. The collision distance, in meters, can be estimated for
equal sized spheres from 1 x 10-5δ0.23d1 for separations of 5 diameters and less, where
δ is the separation in diameters and d1 is the leading sphere diameter. Similarly, the
collision time in seconds for equal sized spheres at δ = 5 or less, can be estimated from
1 x 10-5δ0.18d1/U, where U is the start velocity of the spheres. For unequal sized sphere
pairs, the decreased drag on the smaller trailing sphere is counteracted by faster loss of
momentum compared to the leading sphere. For cases where the ratio of drag
coefficients for the trailing sphere divided by the leading sphere is greater than the
ratio of their diameters, there is less drag improvement from the separation distance
than there is relative deceleration increase from the differences in mass, and the
spheres will not collide. At separations less than six diameters, drag reduction was
found to be significant for both the leading and trailing spheres compared to a single
sphere at the same Reynolds number. At the closest separation of 1.5 diameters, over
10 % drag reduction was found for the leading sphere and up to 50 % drag reduction
was found for the trailing sphere. The primary contributor to the drag reduction on the
trailing sphere was found to be the reduced velocity field created in front of the
trailing sphere. The leading sphere has drag reduction caused by modification to its
wake region. The average drag reduction for both spheres was also found to be within
± 3 % of the creeping flow analytical two sphere solution.
©Copyright by Varna Lindqvist
September 13, 2011
All Rights Reserved
Simulation of Drag Reduction due to Multi-Aligned Spheres
by
Varna Lindqvist
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Presented September 13, 2011
Commencement June 2012
Master of Science thesis of Varna Lindqvist presented on September 13, 2011
APPROVED:
Co-Major Professor, representing Mechanical Engineering
Co-Major Professor, representing Mechanical Engineering
Head of the School of Mechanical, Industrial, and Manufacturing Engineering
Dean of the Graduate School
I understand that my thesis will become part of the permanent collection of Oregon
State University libraries. My signature below authorizes release of my thesis to any
reader upon request.
Varna Lindqvist, Author
TABLE OF CONTENTS
Page
1 Introduction ................................................................................................................. 1
2 Literature Review and Test Plan ................................................................................. 5
2.1 Single Sphere Drag .............................................................................................. 6
2.2 Multiple Sphere Drag .......................................................................................... 8
2.3 Ellipsoid Drag Approximation .......................................................................... 11
2.4 Spherical Approximation for Droplets .............................................................. 11
2.5 Transient Drag Calculations .............................................................................. 12
2.6 Model Formulation and Test Plan ..................................................................... 13
3 Modeling Method ...................................................................................................... 19
3.1 Physical Model .................................................................................................. 20
3.2 Computational Mesh .......................................................................................... 22
3.3 Computational Model ........................................................................................ 27
3.4 Model Convergence and Verification ................................................................ 33
3.5 Validation .......................................................................................................... 36
4 Data Reduction and Analysis .................................................................................... 38
TABLE OF CONTENTS (Continued)
5 Results and Discussion .............................................................................................. 44
5.1 Drag Coefficient Evaluation .............................................................................. 44
5.2 Surface Force Distributions ............................................................................... 55
5.3 Drag Reduction Characterization ...................................................................... 67
5.4 Transient Drag Calculation ................................................................................ 77
6 Conclusion ................................................................................................................. 89
Bibliography ................................................................................................................. 92
Appendices ................................................................................................................... 94
Appendix A – GCI Calculation from Roache [23] ................................................. 95
Appendix B – Table 5: Calculated two sphere drag from simulations ................... 96
Appendix C – Standard error of the fit calculation for power series ...................... 99
LIST OF FIGURES
Figure
Page
1:
Images of ink droplet formation in 5 μs intervals by Lindqvist from
unpublished work........................................................................................……2
2:
Optical image showing spherical droplet shape by Lindqvist from
unpublished work ………………………………………………………...…..15
3:
Schematic of model for 2 aligned spheres…………………………...……… 17
4:
Physical model geometry …………………………...………………………. 20
5:
Single O-grid geometry (a) nodes on inner and outer cubes (b) sphere
mesh with inner and outer cubes………...…………..………………………. 23
6:
Single O-grid geometry (a) in domain (b) zoomed in view of outer cube
with sphere mesh………………………………………………………….…. 23
7:
Mesh elements on plane through sphere center (a) 0.6 μm grid (b) 2.4 μm
grid showing flat surfaces of the mesh.…………..……..…………..………. 24
8:
Two sphere O-grid construction (a) in domain (b) close view.…………..…. 25
9:
Mesh elements on plane through following sphere center, 0.6 μm grid.…..... 26
10:
Mesh discretization locations.…………..……..………………………….…. 28
11:
RMS residuals for two aligned 20μm diameter spheres at 10 m/s and a
separation of 2 diameters….……..……..…………………………...………. 34
12:
Modeled sphere drag coefficients with Oseen [3], Stokes [1] and White [2]
equations..…………..……..…………………………………………...……. 37
13:
Drag coefficients for two aligned 20μm diameter spheres a) leading
b) trailing as a function of separation distance, δ.………..……..……...……. 45
14:
Drag coefficients for two aligned 20μm diameter spheres versus Reynolds
number with single sphere predictions from White [2]: (a) leading
(b) trailing………………………………...…………………………………. 46
LIST OF FIGURES (Continued)
Figure
Page
15:
Drag coefficients for two aligned 20μm diameter spheres versus Re for two
separate distances (a) δ=1.5 (b) δ=5..…………..……..……………………... 47
16:
Drag reduction parameter as a function of separation distance for two
20μm spheres (a) leading (b) trailing..…………..……..……………………. 48
17:
Drag reduction parameters for two aligned 20μm spheres at 10 m/s:
leading and trailing..…………..……..………………………………………. 50
18:
Inertia and geometric contributions to drag reduction parameter λ…………. 51
19:
Drag reduction parameter for two aligned 20μm diameter spheres as a
function of δ and velocities of (a) 6 m/s (b) 14 m/s. .…………..……..……. 52
20:
Drag reduction parameter for two aligned spheres, d1=26 μm, at 10 m/s
and as a function of δ for different size ratios β: (a) leading (b) trailing..….. 53
21:
Drag reduction parameters for two aligned spheres, d1=26 μm, at 10 m/s
as a function of and for β of 0.62, 0.77, and 1..……...…..…………….……. 54
22:
Drag reduction parameter versus sphere spacing for all test cases in Table 2.
For λ2 values of λ are lower with smaller β, higher U and unchanged by d1....55
23:
Leading sphere surface distributions at 6 m/s for d1=26μm and δ=1.5
(a) pressure (b) shear strain..……………………..……….…………………. 57
24:
Trailing sphere surface pressure and shear strain distributions at 6 m/s for
d1=26μm and δ=1.5 (a) pressure (b) shear strain.……….…..………………. 57
25:
Single sphere surface pressure and shear strain distributions at 6 m/s and
δ=1.5 (a) pressure (b) shear strain..…………..……..………………………. 58
26:
Velocity contours for 6 m/s, d1=26 μm, δ=1.5 (a) β=0.62 (b) β=1..…..……. 59
27:
Shear strain contours for 6 m/s, d1=26 μm, δ=1.5 (a) β=0.62 (b) β=1..….…. 60
28:
Pressure contours at 6 m/s, d1=26 μm, δ=1.5 (a) β=0.62 (b) β=1..…………. 60
29:
Velocity vectors for 6 m/s, d1=26 μm, δ=1.5 and β =1 (a) both spheres
(b) zoomed in view..…………..……..…………………….…..……………. 61
LIST OF FIGURES (Continued)
Figure
Page
30:
Leading sphere surface distributions at 6 m/s for d1=26μm and δ=5
(a) pressure (b) shear strain..……………………..……..…...………………. 62
31:
Trailing sphere surface pressure and shear strain distributions at 6 m/s, δ=5
(a) pressure (b) shear strain..………….…..…………………………………. 63
32:
Single sphere surface pressure and shear strain distributions at 6 m/s and
δ=5 (a) pressure (b) shear strain..…………..……..…………………..…..…. 64
33:
Velocity contours for 6 m/s, d1=26 μm, δ=5, β=1.……….…..……..………. 65
34:
Contours for 6 m/s, d1=26 μm, δ=5, β=1 (a) pressure (b) shear strain..…..…. 65
35:
Drag reduction parameter versus sphere spacing with ellipsoid predictions
from Clift[17]..…………..……..……………………………………………. 66
36:
Trailing sphere drag reduction versus δ for equal sized spheres with
equation (54) for δ=11.…………..……..………………………….……..…. 68
37:
Trailing sphere drag reduction versus for equal sized spheres for (a) δ=1.5
(b) δ=5.…………..……..……………………………………………..……. 69
38:
Equal sized trailing spheres values of I2 versus intersphere distance δ-1..….. 70
39:
Trailing sphere intercept values, I2, versus distance (δ-1) for different
sphere size ratios β. …………..…………………………….…………….…. 71
40:
Trailing sphere Φ2 values versus 1/(δ-1) with associated linear curve fits...... 72
41:
Slope, M, of I2 versus 1/(δ-1) for a range of β..…………..……..………..…. 73
42:
Trailing sphere intercept values, I2, with β adjustment factor, Φ2, versus
distance (δ-1) for different sphere size ratios β..…………..……..……….…. 74
43:
Leading sphere drag reduction for equal sized spheres versus intersphere
distance (δ-1)……………………...…………..……..………………………. 75
LIST OF FIGURES (Continued)
Figure
Page
44:
Leading sphere drag reduction, λ1, for all sized spheres versus intersphere
distance (δ-1)..…………………………….…..……..………………………. 76
45:
Leading sphere drag reduction normalized by Φ1 for all sized spheres versus
intersphere distance (δ-1)..…………………....……..………………………. 77
46:
Predicted model output d1=20, β=1, both with initial velocities of 10 m/s,
and δ=4..…………..……..…………………………………………………. 79
47:
Equal sized spheres collision distance xc versus δ for both d1 and all U
(a) d1 labeled (b) U labeled.…………..……..………………………………. 80
48:
Equal sized spheres non-dimensional collision distance ηc to Re.…….….…. 81
49:
Equal sized spheres non-dimensional collision distance ηc by δ to Re..….…. 82
50:
Equal sized spheres collision time tc versus δ for both d1 and all U (a) d1
labeled (b) U labeled.…………..…………………....………………………. 83
51:
Equal sized spheres non-dimensional collision time, τc , to Re.………….…. 84
52:
Equal sized spheres non-dimensional collision time, τc , by δ to Re.……….. 84
53:
β = 0.8 spheres at 10 m/s just converging with δ = 2.695 (a) trajectory
(b) collision criteria.…………..……..………………………………....……. 86
54:
β = 0.8 spheres at 10 m/s just diverging with δ = 2.700 (a) trajectory
(b) collision criteria..…………..……..…………………………………..….. 87
55:
Maximum distance for collision, δc, for smaller trailing sphere versus Re…. 88
LIST OF TABLES
Table
Page
1:
Drag reduction parameter λ of two spheres with center-to-center separation δ
in sphere diameters from Stimson [7] and Faxen [9], tabulated by Happel
[8]…………………………………………………………………………..…..9
2:
Experimental test conditions for 2 aligned spheres ……………………….....17
3:
Additional experimental test conditions for 2 aligned spheres ………...…….18
4:
Modeled sphere drag coefficients with errors from Oseen [3], Stokes [1]
and White [2].…………………………………………………….…………..37
Nomenclature
a
Sphere radius (μm)
As
Frontal area (m2)
b
Ellipsoid short axis length (m)
c
Ellipsoid long axis length (m)
Cd
Drag coefficient
d
Sphere diameter (μm)
g
Gravitational acceleration (m/s2 )
F
Force (N)
Fg
Gravitational force (N)
I2
Trailing Sphere Re fit intercepts
m
Mass (kg)
M
Slope in equation for Φ
ng
Nanograms
p
Pressure (Pa)
Re
Reynolds number Re 
S
Trailing sphere β fit slope
t
Time (s)
U
Constant freestream velocity (m/s)
V
Velocity (m/s)
V
Volume (m3)
X, x
Position (m)
Ud

Nomenclature (continued)
Greek Symbols
α
Acosh δ
β
Sphere size ratio
δ
Center to center separation distance in diameters based on the leading sphere
diameter
t
Timestep (s)
λ
Drag reduction parameter
μ
Dynamic viscosity (Pa·s)
ν
Kinematic viscosity (m2/s)
ρ
Density (kg/m3)
σ
Surface tension of water (mN/m)
σ
Stability parameter
τ
Shear Stress (Pa)
τ/μ
Shear strain rate (1/s)

Transported quantity
Φ
β adjustment factor
Nomenclature (continued)
Subscripts
1
Leading sphere of two
2
Following sphere of two
c
Collision
g
Gas
I
Initial
l
Liquid
n
Current timestep
n-1
Previous timestep
Superscripts
*
Non-dimensional form
Drag Reduction due to Multi-Aligned Spheres
1 - Introduction
Thermal inkjet micro-fluidic devices generally produce a larger main droplet
followed by one to five smaller satellite droplets. With a single ink droplet it is
straightforward to predict the flight path of the droplet between the inkjet cartridge and
the paper. The droplet can be approximated by a sphere and the known drag
coefficients for a sphere used to update the drag force on the droplet as it slows over
time due to friction and pressure from the air. Once the flight path is known the
location of the single droplet on the paper can be predicted, which is essential to
produce crisp images and text. It would, therefore, be useful to be able to predict the
locations on paper of all the droplets in a realistic multiple droplet stream. A method
does not exist to estimate the drag coefficients for multiple droplets at close spacings
where the drag coefficients will likely deviate significantly from the single sphere drag
coefficient values. Thus, to estimate the flight paths for multiple droplets accurately,
the separate drag coefficients for all droplets in the chain need to be determined. The
usual geometries of inkjet droplets are quite complex and indeed the droplets are
closely spaced, as shown in the time series photographs of the formation of ink
droplets in Figure 1. Due to the complexity of the droplets, gathering reliable
experimental measurements of the velocities of each droplet over a range of distances
from the inkjet cartridge to quantify the drag behavior would be quite difficult. To
simplify from the observed droplet behavior two spheres could be used to approximate
most of the drag difference between multiple droplets. For two closely spaced spheres
2
inline with the air flow, it is thought that the leading sphere would shield the trailing
sphere from much of the air flow, resulting in lower drag on the trailing sphere.
Additional trailing spheres would be similarly shielded and likely have drag about the
same as that of the first trailing sphere.
Figure 1: Images of ink droplet formation in 5 μs intervals by Lindqvist from
unpublished work.
There is no known analytical solution to accurately predict the drag
coefficients for a single sphere for the range of sphere diameters and velocities
3
observed from inkjet droplets. Therefore, a two sphere analytical solution is also not
possible. Drag coefficients for a single sphere in the flow regimes typical for inkjet
droplets have been reliably estimated through numerical simulation using
computational fluid dynamics (CFD) software programs. Simulations with a single
sphere model will be run first and the output compared to an accurate fit to
experimental data to confirm that accurate drag coefficients are being obtained from
the simulations. Constant velocity air will be applied to two fixed spheres a
designated separation distance apart and the simulations run until a stable drag force
distribution develops around the spheres. These distributions from the pressure and
viscous forces on the spheres are then integrated around the spheres to obtain the total
drag force on each sphere. The forces will be put into the form of drag coefficients so
that they are applicable for different sphere sizes and velocities beyond those in the
models. Equations will then be sought to describe these drag coefficients as a function
of the separation distance between the spheres. The drag will likely be different for
the two spheres so separate equations will be found for the leading and trailing
spheres. Once these drag equations are determined they will be incorporated into a
calculator to find the distances traveled by the leading and trailing spheres. Initial
sphere velocities and separation distance will be input into the calculator. Drag forces
from the equations for these initial settings will be applied for a small increment of
time then new slower velocities and the position changes for the two spheres will be
calculated. The drag coefficients will be updated for these new velocities at the
separation distance and the corresponding drag force applied for another small
4
increment of time. These iterations will be performed by the calculator until a
specified amount of time has passed or the two spheres collide. The distances
traveled, velocities and relative separation of the two spheres will be output by the
calculator for each time increment. Updating the drag coefficients separately from the
drag equations for the leading and trailing spheres will provide better trajectory
estimates, than could be obtained from using the single spheres drag coefficients for
each of the spheres.
5
2 Literature Review and Test Plan
The velocity of an object moving through a gaseous or liquid fluid is retarded
by drag forces from friction as the fluid slows to zero velocity on the object surface
and from the pressure difference on the object’s surface as the fluid is pushed out of
the way and fills in behind the object. These viscous and pressure forces sum to the
total drag force, F, given by:
F 
C d ρU 2 As
2
(1)
where ρ is the fluid density, U is the constant relative velocity between the fluid and
object, and the frontal area, As, which for a sphere is πa2. The drag coefficient, Cd,
must be found experimentally or through computer simulation for most flows and is in
general a function of the objects’ Reynolds number. An object’s velocity is also
generally modified by the gravitational force, Fg, given by:
Fg  mg
(2)
However, for very small masses at high velocities the effect of gravity can be
neglected.
Discussed in the following section are single sphere drag solutions which are
used to validate the modeling methodology and in the two sphere drag evaluation.
Multiple sphere drag solutions, drag on ellipsoidal shapes and the appropriateness of
the spherical approximation for droplets are then reviewed. Lastly, the model
formulation and test plan are presented.
6
2.1 Single Sphere Drag
The drag behavior of one sphere is well known, as discussed later, and
provides an upper bound on the drag coefficients that should be obtained for two
aligned spheres. Analytical solutions are only possible through simplifications of the
full Navier-Stokes momentum equation and result in simple, though limited, equations
to estimate the drag coefficients. The Navier-Stokes momentum equation for constant
viscosity, μ, constant density, ρ, flow where gravity is negligible is given by:

DV
 p   2V
Dt
(3)
Stokes [1] postulated that the inertia terms in the momentum equation (3)
could be neglected for very slow “creeping flows”. An order of magnitude analysis of
the dimensionless momentum equation, scaling pressure with the viscous scale, μU/L,
gives:
Re
DV *
  * p *  *2 V *
Dt *
(4)
where:
Re 
Ud

(5)
for a sphere of diameter d. Since the right hand side terms are of the order of 1, the
inertia term on the left hand side can be neglected for Reynolds number, Re, much less
than one, leaving the Stokes [1] assumption of:
p   2V
(6)
7
Stokes [1] solved this reduced momentum equation directly to yield the sphere drag
formula:
F  6aU
(7)
Two-thirds of the force F in equation (7) is viscous force from integration of the shear
stress distribution around the sphere. The remaining one-third is pressure force from
integration of the pressure distribution. The drag coefficient based on Stokes [1]
results is then:
Cd 
2F
24

2
2
Re
ρU πa
(8)
From White [2], equation (8) “is strictly valid only for Re << 1 but agrees with
experiment up to about Re = 1”.
Oseen [3] added to equation (6) a linearized simplification of the full inertia
term by substituting the constant freestream velocity, U, for the u, v, and w velocity
components in the inertia term to give:
U
V
 p   2V
x
(9)
which modifies the Cd to be:
Cd 
24 
3

1

Re

Re  16 
(10)
At large distances from the sphere, Oseen’s [3] inertia term seems appropriate as the
fluid velocity will be close to the freestream velocity. However, the no-slip boundary
condition requires zero velocity on the sphere surface, so use of the freestream
8
velocity here and in the neighborhood around the sphere appears to be a poor
approximation of the true inertial term [2].
An analytical solution has not been developed to predict drag coefficients
accurately for Re > 1. The best estimates of Cd values are from fits to experimental
data and numerical simulations. A fit to the experimental sphere Cd data by White [2]
gives the following formula, which is accurate within ±10% for Re up to 2 x 105:
Cd 
24
6

 0.4
Re 1  Re
(11)
Subsequent numerical studies from LeClair [4], Dennis [5] and Feng [6] agree with
the Cd values from White [2] within ±3% for Re 1, 5, 20 and 40. The White [2] fit,
equation (11), is thus judged to be an accurate predictor of the drag coefficients for a
single sphere in the Re 5 to 40 range, which is to be modeled in the present study.
Simulations will be made for models with a single sphere and the results compared to
White’s [2] fit, equation (11), to validate the modeling procedure.
2.2 Multiple Sphere Drag
Stimson [7] developed an analytical aligned two sphere solution “based on
determining Stokes stream function for the motion of the fluid, and from this the
forces necessary to maintain the motion of the spheres” [8]. For two equal sized
spheres moving parallel to their line of centers the solution gives the drag force as:
F  6aU
(12)
where:
4
3
 4 sinh 2 (n  1 / 2)  (2n  1) 2 sinh 2  
n(n  1)
1 

2 sinh(2n  1)  (2n  1) sinh 2 
n 1 (2n  1)(2n  3) 

  sinh  
(13)
9
and  is given by:
cosh   
for δ >1
(14)
The variable δ is the center-to-center separation distance of the two equal sized
spheres in sphere diameters. For the limiting case where the spheres touch at δ =1,
λ was calculated by Faxen [9] as:
lim  
 0

4 1  4 sinh 2 x  4 x 2 
1 
dx
3 0 4 
2 sinh 2 x  4 x 
for δ =1
(15)
The value of λ is between 0 and 1 and as such is identified as the drag reduction
parameter. Equation (12) shows that λ reduces the drag force compared to the
Stokes[1] solution for a single sphere given by equation (7). Values of λ from
equations (13) for δ >1 and (15) for δ =1 tabulated by Happel [8] are given in Table 1.
Table 1: Drag reduction parameter λ of two spheres with center-to-center separation δ
in sphere diameters from Stimson [7] and Faxen [9], tabulated by Happel [8].
α
0
0.5
1
1.5
2
2.5
3
δ
λ Happel [8]
1
1.128
1.543
2.352
3.762
6.132
10.068
0.645
0.660
0.702
0.768
0.836
0.892
0.931
The values of λ are reported to be the same for both spheres, since following Stokes
[1] the inertia term is neglected and the remaining viscous and pressure terms are the
same for equal sized spheres. In creeping flow the pressure fields from the leading
and trailing spheres influence each other equally, resulting in streamlines that are
symmetric about a plane midway between the spheres, and thus equal drag reduction
10
occurs for the two spheres. The solution, to equations (13) and (15), has excellent
agreement with the experimental data from Bart [10] for two equal sized spheres
falling aligned along their centers in a cylinder 61 times wider than the diameter of the
spheres at Re = 0.05 [8]. Happel [8] used a different analytical technique known as
the method of reflections, to yield the same λ values as in Table 1 to three significant
digits. A later solution by Gluckman [11] confirmed the two sphere λ values and
extended a solution methodology to predict drag coefficients for chains of 3 to 7
aligned spheres.
From White [2], “The streamlines (for creeping flows) possess perfect foreand-aft symmetry. It is the role of the convective acceleration (inertia) terms to
provide the strong flow asymmetry typical of higher Reynolds number flows”. The
character of the asymmetry is that the streamlines, i.e. lines of constant velocity, are
deflected further outward on the trailing versus the leading side of the sphere.
Practically, this means that for a given sphere, as the freestream velocity increases
beyond Re = 1, a progressively larger region with a lower velocity than the freestream
develops on the trailing side of the spheres. For the two spheres to be modeled, at
distances where the following sphere is within this lower velocity region from the
leading sphere, there will be less drag force on the following sphere compared to the
leading sphere. For two aligned spheres at the same initial velocity, with less drag
force the following sphere will continually move closer to the leading sphere. Indeed
this effect has been noted even at very low Re numbers. As reported by Happel [8]
“When two spheres fall at Reynolds numbers over 0.25, inertial effects are
11
experimentally noted, in that the spheres no longer maintain a fixed position relative to
each other as they do in the creeping motion regime”. From this inertial effects exist
even for very low Re flows.
2.3 Ellipsoid Drag Approximation
It is possible, that between the two modeled spheres at close distances, there
will be a very low velocity region, resulting in drag similar to a solid ellipsoid with a
length the same as the distance from the front of the leading sphere to the back of the
trailing sphere. An approximate formula from Clift [12] for the drag ratio of an
ellipsoid along the long axis, to a sphere the diameter of the short axis, is given by:
c

4 
b
E  
 5 




(16)
This equation agrees with the exact creeping flow solution for an ellipsoid in a prolate
orientation to within ± 1 % error for 0 < c/b < 5, where c is the long axis length and b
is the short axis length of the ellipsoid.
2.4 Spherical Approximation for Droplets
Using a rigid sphere in place of water droplets is desirable as it greatly
simplifies the fluidic modeling. However, to ensure accurate results the
appropriateness of the spherical approximation must be established. An analytical
solution of a spherical liquid drop in creeping flow developed independently by
Hadamard [13] and Rybczyski [14] is:
F  6a gU
1  2 g / 3 l
1   g / l
(17)
12
where, μl, is the liquid viscosity and, μg, the gas viscosity. Water droplets have
viscosity 56 times greater than air viscosity. Thus, the added term becomes 0.994,
which is very close to 1 where Stokes [1] law is recovered, indicating that a water
droplet in air has very nearly the same drag force as a rigid sphere in air. Equation
(17) is a creeping flow solution and so may apply for Re < 1. However, a water
droplet would likely become non-spherical with a large diameter at high velocities.
Beard [15] reported from experiments that "for water drops in air, a plot of Cd versus
Re follows closely the curve for rigid spheres up a Reynolds number of 200,
corresponding to a particle diameter of 0.85 mm". A parameter often used to account
for surface tension, σ, effects which may influence droplet shapes, is the Weber
number:
We 
U 2 d

(19)
Warnica [16] found that for Re = 10, Weber numbers no larger than 0.0466 resulted in
Cd values that deviated less than 1% from those for a rigid sphere.
2.5 Transient Drag Calculations
Michaelides [17] performed calculations which determined that neglecting the
transient terms of the force, for a 100 μm diameter water droplet in an unsteady air
velocity field, resulted in error in the total distance traveled of less than 4% even at
fluid frequencies up to 81 Hz. Michaelides [17] also calculated the transient velocity
for a particle in air using an empirical correlation from Rowe [18], similar to equation
(11) from White [2], for the steady state drag coefficient:
13
Cd 

24
1  0.15 Re 0.667
Re

(18)
Using an integration of this equation over a small timestep with Re as a constant,
resulted in error in the instantaneous velocity of more than 5% for a timestep to total
time ratio of 0.02 compared to values from the exact integral [17]. At each timestep
the Re value was updated and the error in the instantaneous velocity decayed
exponentially with added timesteps to near zero at the end of the calculation.
Comparison with the 0.1 timestep to total time ratio calculation shows a corresponding
5 to 1 ratio in the instantaneous velocity error over the calculation time period
indicating that the error decreases linearly with timestep, so with small timesteps
correspondingly small errors in the instantaneous velocity should be obtained with this
iterative technique.
2.6 Model Formulation and Test Plan
There is no known equation to predict the drag coefficients for two closely
spaced spheres or droplets for the Re = 5 to 40 range, so a fluidic model will be
developed and simulations run to generate the associated drag coefficients. For ease
of modeling, several simplifications of the physics for the drag forces on ink droplets
are made. The simplifications include neglecting body forces, substituting rigid
spheres in place of droplets, reversing the frame of reference to be on the spheres, and
obtaining steady state solutions then computationally determining the transient
behavior with an iterative technique instead of using transient simulations. These
simplifications are justified below.
14
2.6.1 Physical Simplifications and Model Formulation
Gravitational and buoyancy forces are proportional to the mass of an object.
The largest ink droplet to be modeled has a 9.2 x 10-12 kg mass which at the 1000
kg/m3 density of water corresponds to a 26 μm diameter sphere. Ink droplets typically
have a velocity of about 10 m/s. Using equation (2) with this mass and the
gravitational acceleration of 9.8 m/s2 and equation (1) with U = 10 m/s, a = 13 μm, air
density ρ of 1.185 kg/m3 and the Cd from White’s [2] equation (11) for a single sphere,
a water droplet moving downwards is influenced by a gravitational force only 0.07 %
of the 1.3 x 10-7 N drag force. Thus, the mass of the droplets and the associated
gravitational and buoyancy forces can be neglected.
From equation (19) where σ is the surface tension of water, 72.8 mN/m at
200C, the largest spheres to be modeled, 26 μm in diameter, have We numbers of
0.00002 at the highest velocity of 22 m/s, which is only 0.04% of the 0.0466 limit
recommended by Warnica [16]. This modeled sphere We number is also only 11% of
the limit for water droplets reported by Beard [15]. Since We numbers for the water
droplets to be modeled are much less than the We numbers for droplets that have been
observed to be spherical, the use of rigid spheres in the models is a valid
simplification. The spherical shape of droplets approximately the same size as those to
be modeled is also confirmed with optical images as shown in Figure 2.
15
Figure 2: Optical image showing spherical droplet shape by Lindqvist from
unpublished work.
Ink droplets typically move at velocities through otherwise near quiescent air.
In the models the frame of reference is to be reversed. Rigid spheres representing ink
droplets will be stationary and a constant air velocity applied to them. Since, the
velocity of the droplets or air were both to be constant, changing the frame of
reference is transparent in regards to the forces to be developed on the sphere surfaces.
Furthermore, since droplet mass is neglected the spheres will be fixed in place so they
cannot drift.
Evaluating transient simulations for two spheres would be complicated by
trying to separate the transients due to the developing velocity field around the spheres
from the transient motion of the spheres as they slow under the drag from the velocity
field. Steady state simulations avoid this difficulty by running to long time periods
compared to the transience in the developing velocity field, thus retaining only the
steady state drag solution. Once the steady state drag forces are determined from the
simulations for a range of conditions, empirical equations will be found to calculate
the drag coefficients as a function of the separation distance between the two spheres.
These drag coefficient equations will then be used to iteratively determine the
velocities and positions of two spheres of specified size, initial velocity and
16
separation. Using an iterative technique to determine the transient velocity from an
empirically determined equation for the drag coefficients has been shown to have
errors of 6% or less for a timestep to total time ratio of 0.02 [17]. Using the velocity
error to timestep relationship from Michaelides’ [17] calculations, the 0.0025 ratio of
the 5 x 10-7 s timestep used to ensure stability to the estimated 2 x 10-4 total time for
droplets to reach the paper, predicts instantaneous velocity errors of less than 1%. The
deceleration of the droplets is not to be used to modify the velocity in this iterative
technique. The error from not including these transient forces is thought to be less
than 5% from calculations done by Michaelides [17] for similar water droplets in air.
2.6.2 Test Plan
To achieve good resolution for cases where the drag reduction is expected to
be most substantial, the test plan focuses on separation distances equal to five
diameters and less. Simulations for the 20 μm equal sized sphere case at the middle
velocity setting of 10 m/s are to be run at separation distances up to 19 diameters, in 1
diameter increments to find the separation distance that causes the drag reduction to go
to zero for two aligned spheres.
The models are to consist of two aligned spheres with sizes d1 for the leading
sphere diameter and d2 for the following sphere diameter at separation distances, δd1,
of 1.5, 2, 3, 4, and 5, as shown in Figure 3.
17
β = d2/d1
Spheres fixed in space
d1
d2
Freestream
velocity U
δd1
Figure 3: Schematic of model for 2 aligned spheres
Freestream velocities, U, of 6 m/s, 10 m/s and 14 m/s are to be applied to models at
each of the separation distances, for the cases shown in the simulation plan in Table 2.
Table 2: Experimental test conditions for 2 aligned spheres
U [m/s] d1 [μm] d2 [μm]
6
10
14
6
10
14
6
10
14
6
10
14
6
10
14
20
20
20
20
20
20
26
26
26
26
26
26
26
26
26
16
16
16
20
20
20
16
16
16
20
20
20
26
26
26
δ [d1]
β
0.80
0.80
0.80
1.00
1.00
1.00
0.62
0.62
0.62
0.77
0.77
0.77
1.00
1.00
1.00
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
Re1
Re2
7.7
12.9
18.1
7.7
12.9
18.1
10.1
16.8
23.5
10.1
16.8
23.5
10.1
16.8
23.5
6.2
10.3
14.5
7.7
12.9
18.1
6.2
10.3
14.5
7.7
12.9
18.1
10.1
16.8
23.5
18
Calculated Re1 and Re2 values, for d1 and d2 spheres, respectively, based on U are
included to show the Re range. The 16 μm, 20 μm and 26 μm sphere diameters in the
models correspond to representative inkjet droplet weights of 2.1 ng, 4.2 ng and 9.2
ng, respectively. The ratio of the trailing sphere diameter, d2, divided by the leading
sphere diameter, d1, is given by β. A β of 1 is considered to be a reasonable upper
limit, as ink droplets have not been observed with trailing droplets larger than the
leading droplet. To represent geometries often seen in ink droplets, sphere pairs with
β values of 0.62, 0.77 and 0.80 are also to be modeled in addition to the β = 1 case.
For the 20 μm equal sized sphere case, separation distances from δ = 6 to δ = 19 in
δ = 1 increments are to be modeled to resolve the full extent of the drag curves for
both spheres. Also for the 20 μm equal sized sphere case, additional models with
intermediate velocities of 8 m/s and 12 m/s and are to be created to provide well
defined curves versus Reynolds number, and additional models with intermediate δ
spacing of 2.5 and 3.5 are to be created to show curvature versus δ. These additional
cases are shown in Table 3.
Table 3: Additional experimental test conditions for 2 aligned spheres
U [m/s] d1 [μm] d2 [μm]
8
12
20
20
20
20
10
20
20
β
1.00 1.5 2
1.00 1.5 2
2.5 3.5
9
10
1.00
14 15
19
δ [d1]
3
3
6
11
16
Re1
4
4
7
12
17
Re2
5 10.3 10.3
5 15.5 15.5
8
13
12.9 12.9
18
19
3 Modeling Method
To estimate the velocities of two closely spaced ink droplets, the drag
coefficients of both droplets as a function of their separation distance needs to be
determined. Computational fluid dynamics (CFD) is employed to simulate the drag
forces on both the leading and trailing droplets. In the CFD model, air at constant
velocity, U, is applied to two stationary spheres within a large domain and the solver
run until a steady state solution is obtained. Applied velocity, sphere diameters and
separation distance between the spheres are varied to simulate drag over the estimated
range of interest. From the forces on the spheres, drag coefficients can be calculated
and the continuous drag behavior for each sphere described by a curve fit of the Cd
data at the discrete distances that are modeled. Numerical CFD simulations were
undertaken using the CFX and ICEM codes, which are components of the
commercially available software package Ansys ®, version 12.1. The CFX software
was used to set up the models with appropriate values and properties, solve the
simulations and postprocess the results for the desired outputs. The meshes used by
the CFX program were developed in ICEM and imported into CFX. The Cartesian
coordinate system is used in both ICEM and CFX for the model and mesh geometries,
solver parameters, residual calculations and output forces. Discussed in the following
sections are the physical model constructed, the computational mesh developed, the
computational model, the model convergence and verification, and validation against
known experimental data.
20
3.1 Physical Model
The model geometry was built in ICEM and consists of one or two solid
spheres within a three dimensional cube shaped domain. A two dimensional
representation of the geometry is shown in Figure 4.
b.c. free-slip walls
β = d2/d1
Spheres fixed in space b.c.
no-slip walls
b.c. inlet
d1
d2
b.c. pressure
outlet
δd1
+y
Constant inlet
velocity
Fluid at constant
initial velocity
+x
b.c. free-slip walls
Note: Not to scale
Figure 4: Physical model geometry
An inlet boundary conditions (b.c.) is set at the entrance and a pressure outlet b.c. is
set at the exit of the domain. The sides of the domain are specified as free-slip walls
and the spheres are given a no-slip boundary condition. Sphere diameter is designated
by d for a single sphere and d1 or d2 for the leading or trailing sphere, respectively.
The ratio of the following sphere diameter, d2, divided by the first sphere diameter, d1,
is given by β. The center-to-center separation distance between the two spheres is
21
given by δ in units of diameters based on d1. Sphere sizes are 16 μm, 20 μm and
26 μm in diameter, which provides a representative range for ink droplets. The
spheres are fixed in space and a constant velocity applied at the inlet of the domain in
the +x direction. The entire fluid domain was also initialized with the same velocity
everywhere. A large domain size of 577.2 μm along each edge was employed to
remove any influence from the walls on the solution.
The fluid was specified as air with constant properties at 25OC, with values in
CFX of 1.831 x 10-5 kg/m-s for dynamic viscosity and 1.185 kg/m3 for density. The
reference pressure was set to 101,300 Pa for atmospheric pressure and a relative
pressure of 30 Pa was set at the pressure outlet. At the highest modeled velocity of 22
m/s the Reynolds number for the largest diameter sphere of 26 μm is 36.9. For
external flows the transition to turbulence does not occur until approximately
Re = 2.5 x 105 [2]. The laminar flow model was chosen since the sphere sizes and
velocities modeled are within the laminar regime. The incompressible flow regime
was selected because the highest velocities modeled are less than the 0.3 Mach
number velocity of about 100 m/s for air, where compressible flow is significant. The
energy equation was not solved because it was not needed as no heat is generated from
incompressible flow. For constant density with no source terms the governing
equations to be solved are the mass equation:

U j   0
x j
and momentum equations, both in Cartesian coordinates.
(19)
22

U i    U jU i    P  
t
x j
xi x j


U j
   U i 
eff 

xi
 x j





(20)
Although a steady state condition is specified, the transient term is retained.
The transient term is used by the solver to apply under relaxation parameters to the
pressure and velocity fields to stabilize the solution.
3.2 Computational Mesh
Structured meshes were computed in ICEM for each modeled sphere size and
separation distance articulated in the test plan in Tables 2 and 3. The single sphere
models use a single O-grid to define the mesh geometry. The O-grid is formed from
inner and outer cubes centered in the same location. These cubes have the same
number of nodes along each edge, as shown in Figure 5, which form even grids on the
cubes faces. The inner cube is sized so that the vertices of the cube intersect the
sphere surface. The even grid on the cubes faces is projected radially outward onto
the sphere to form the sphere mesh. The inner cube with sphere mesh is also shown in
Figure 5.
23
(a)
(b)
Figure 5: Single O-grid geometry (a) nodes on inner and outer cubes (b) sphere mesh
with inner and outer cubes.
The sphere mesh extends radially outward to a sphere intersecting the outer cube of
the O-grid and then along vertices to the corners of the domain. Figure 6 shows the
O-grid geometry and outer cube with meshed sphere.
(a)
(b)
Figure 6: Single O-grid geometry (a) in domain (b) zoomed in view of outer cube
with sphere mesh.
24
Beyond the sphere intersecting the outer cube the mesh gradually distorts from the
spherical shape of the outer sphere to the cube shape of the domain. A plane cut
through the sphere center in Figure 7 shows the radially extending elements of the
O-grid mesh. Since the mesh is composed of flat rather than curved surfaces, as
shown in Figure 7, it can be accurately represented by the Cartesian coordinate system
used.
(a)
(b)
Figure 7: Mesh elements on plane through sphere center (a) 0.6 μm grid (b) 2.4 μm
grid showing flat surfaces of the mesh.
The O-grid structure creates a gradual decrease in refinement with distance away from
the sphere surface. The grid elements are highly refined near the sphere surface to
resolve the high velocity, shear and pressure gradients close to the sphere. Sufficiently
fine meshes are necessary in areas with high gradients to minimize discretization
errors, which is one requirement to obtain accurate computational solutions. Grid
spacing on the sphere surface was set to 0.6 μm for the 20 μm diameter sphere. The
25
grid spacing was scaled with sphere diameter to create the same level of refinement on
the 16 μm, 20 μm and 26 μm diameter sphere models.
For the two sphere models, shown in Figure 8, there is an O-grid around each
sphere which is joined to a block between the two spheres.
(a)
(b)
Figure 8: Two sphere O-grid construction (a) in domain (b) close view
The between spheres rectangular shaped block in Figure 8 defines a third O-grid mesh
which is a cylinder rather that a true sphere. The circular face of the cylindrical mesh
is in the yz plane and the length of the cylinder is in the x plane. Figure 9 shows the
mesh around the trailing sphere where the O-grids are joined.
26
Cylindrical O-grid
join
Spherical O-grid
20% length increase per cell
Figure 9: Mesh elements on plane through following sphere center, 0.6 μm grid
To avoid discontinuities in the forces on the sphere surface, elements in the spherical
and cylindrical O-grids are sized similarly. The cylindrical O-grid between the
spheres was set to a refinement of 0.6 μm on the surface of each sphere and the cell
thickness set to increase by 20% for each successive node away from the sphere
surface. The number of nodes in the cylindrical O-grid between the two spheres was
varied to accommodate the sphere spacing difference while maintaining the 0.6 μm
resolution at the sphere surfaces.
The mesh is formed in a structured manner with all nodes and refinement in
specified locations defined by the number and spacing of nodes set on the O-grid
cubes and the geometry of the cubes and vertices extending through the domain. The
27
mesh is then converted into an unstructured mesh since CFX is an unstructured solver.
The information in the mesh file contains a number and cartesian coordinates for each
node and a list of the node numbers which are neighbors to that node. The coordinates
and neighbors are used by the solver directly, rather than mapping nodes onto an even
Cartesian grid as was done by early generation unstructured solvers. With the exact
locations of nodes known an unstructured solver is equal in accuracy to a structured
solver.
3.3 Computational Model
CFX utilizes an element based finite volume method to discretize the
governing equations. The mesh spatially divides the domain and nodes in the mesh
become the corners of volume elements. The centroid of each volume element is
joined with the centroids of adjacent elements to define control volumes, shown by the
area inside the dashed lines in Figure 10. The governing equations (19) and (20) are
integrated over each control volume using the forms given by equations (21) and (22),
respectively.
 U
j
dn j  0
(21)
S
 U
U j

U i dV   U jU i dn j    Pdn j    eff  i 

t V
xi
S
S
S
 x j

dn j


(22)
Advection, diffusion, pressure and mass flux terms are accounted for at the surfaces of
the control volumes, so these terms require surface integrals to be physically
meaningful. To integrate these terms over the surfaces, Gauss’ Divergence theorem is
applied to convert the volume integrals into surface integrals. The time derivatives are
28
also moved outside the integrals, since here the control volumes do not deform with
time. The single remaining volume integral, designated with the subscript V, is the
transient term which sums the mass variation with time inside each control volume.
The differential Cartesian components of the outward normal surface vector are given
by dnj.
The surface integrals are then discretized at the integration points (ip) which
are located inside each element, halfway between the element centroid and the
midway point between two nodes.
nodes
element
element centroid
ip
control volume
midway point between
centroids/nodes
sector
Figure 10: Mesh discretization locations
In these locations, the ip lie on the surfaces between control volumes, so are
automatically locally conservative. The volume integrals are discretized at the sectors
defined inside each element by splitting the element along the surfaces between the
centroid to midway points between nodes.
29
The gradients of the governing equations need appropriate approximations to
estimate the fluxes accurately. Higher order discretization schemes are desirable as
they can lead to lower discretization errors if properly formulated. All solution
variables and fluid properties are defined and evaluated at the mesh nodes. However,
to evaluate the terms the solution gradients must be approximated at the ip of the
control volumes.
The diffusion gradients are discretized using central differencing which is
second order accurate. To achieve an equally weighted linear interpolation between
the upwind and downwind nodes, the diffusion terms are evaluated at the midway
points between nodes. The diffusion term value at the evaluation point is thus the
arithmetic mean of the upwind and downwind nodes. The pressure gradients are
discretized in the same manner. Diffusive properties (here viscosity and pressure)
affect the flow both upstream and downstream, so linearly interpolated values from the
upwind and downwind nodes are physically meaningful approximations.
Advection physically transports mass in the direction of the flow and is thus
primarily influenced by upstream conditions except in recirculating flows. The
simplest discretization for the transported quantity  at the ip is  at the upwind node
which is called upwind differencing (UD). The UD scheme is robust, but only first
order accurate and introduces diffusive discretization errors which lead to poor
resolution of steep spatial gradients.
The advection scheme selected was the high
resolution scheme, which is a UD scheme with a correction term given by: [20]
ip  up    r
(23)
30
The vector r is from the upwind node to the integration point and  is the gradient
in  at the upwind node. The value for  is constrained to be between 0 and 1.
Where  =1 the discretization is second-order accurate and where  =0 the
discretization is first-order accurate. The setting for  is varied throughout the
domain from near 0 for robustness where there are high gradients to near 1 for
accuracy in regions with low gradients. This scheme uses a stencil of adjacent nodes
to compute ß for every node, which ensures boundedness. Details of the ß calculation
are proprietary and not disclosed. The high resolution scheme has been shown to be
total variation diminishing (TVD) in one dimension [20]. Schemes that are TVD have
a total variation of the discrete solution which diminishes with time [21]. Verstreeg
[21] found that all TVD schemes with discretizations using limiter functions, that are
presumably similar to the  calculation, gave “second-order accurate solutions that are
free from non-physical wiggles, so all are suitable for general purpose CFD
computations”.
The transient term is discretized with the Second Order Backward Euler
scheme given by:

1 3
1

dV  V     2 O   OO 

t V
t  2
2

(24)
This scheme is second order accurate and conservative in time though not bounded.
The scheme is also robust, implicit and free of timestep limitations [20].
The correction of a scalar variable  depends on the local velocity field.
However, the velocity field is not known beforehand, but is generated as part of the
31
solution process. The pressure gradient which is linked to velocity in the momentum
equation is also not known beforehand. However, if the pressure gradient was known,
the velocity from the momentum equation could be discretized similarly to the
advection and diffusion terms [21]. The difficulty is resolved by recognizing that “if
the correct pressure field is applied in the momentum equation(s) the resulting velocity
field should satisfy continuity” [21]. Thus, this pressure velocity coupling needs to be
addressed in either the momentum or the mass flow continuity discretizations. The
ANSYS CFX program uses a co-located (non-staggered) grid which stores all values
at the nodes in the center of the control volumes. With this co-located arrangement
the 2nd order central differencing discretization of the pressure term in the momentum
equation using the pressure from the upwind and downwind nodes will not provide an
accurate estimation of the pressure at the present node in a highly varying pressure
field [21]. To correct for this in the interpolation of the face velocities at the midway
point between nodes a 3rd order pressure-correction term from Rhie [22], which
incorporates the pressure at the present node and adjacent nodes, is added to the nodal
velocities output from the momentum equation(s). These face velocities are then used
to discretize the continuity equation, which embeds the pressure corrected velocities
into future iterations. A proprietary modification is also made to the discretization of
the continuity equation, which removes the dependence on the timestep, eliminating
the need to set under relaxation parameters.
The hydrodynamic equations, given by u, v, and w, for x, y and z momentum,
respectively, and p for mass, are solved as a single system using a coupled solver. The
32
linear set of mass and momentum conserving equations resulting from application of
the finite volume discretizations to all elements in the domain is given by: [20]

nbi
ainbinb  bi
(25)
In this form a is the coefficient for each node i, nb represent the neighbors including
the present i node,  is the solution and b is the right hand side (RHS) of the equation.
For the coupled 3D mass-momentum equations ainb is a 4x4 matrix with coefficient
values for all u, v, w and p pair combinations and inb is a 4x1 vector with solution
values for u, v, w and p. An Incomplete Lower Upper (ILU) factorization matrix
inversion technique is used iteratively solve the system of linearized equations. In
matrix form the linear set of equations is:
   b
(26)
The residual between the current and previous solution iteration, r n , is used to obtain
a correction,  ' , which is added to the current solution,  n , to give an improved
solution,  n 1 [20].
r n  b   n
(27)
 '  r n
(28)
 n1   n   '
(29)
The solver will continue to iterate through equations (26), (27), (28) and (29) until the
residual, r n , meets the tolerance that is set or the specified number of outer iterations
is exceeded. Each solution iteration is one outer iteration.
33
The solver convergence is augmented by use of an algebraic multigrid
technique. Iterative solvers efficiently reduce errors of the same order as the mesh
spacing, but errors with longer wavelengths persist for many timesteps. Summing of
the fine mesh discrete equations and their respective control volumes gives a coarse
mesh system of equations which is then solved to reduce higher wavelength errors. In
standard iterative schemes the residual will often decrease rapidly then stall. With this
multigrid acceleration as soon as residual decrease slows iterations are transferred to a
coarser or finer grid, as needed. This allows for a continual decrease in the residuals
and leads to much quicker solution convergence [21].
3.4 Model Convergence and Verification
Residuals are one measure of the degree to which a solution is not exact.
Residuals are the difference in the solution values, u, v, w and P, between the current
and previous timestep. For steady-state solutions each outer iteration is one timestep.
The magnitude of the timestep is automatically set from a physical length scale in the
model, for steady simulations. For steady state solutions the documentation estimates
that 50 to 100 outer iterations are needed for solution convergence [20]. The number
of outer iterations/timesteps was thus set to 100. For steady solutions one inner
iteration to linearize the equations is performed for each outer iteration. In converging
solutions the residuals should decrease with each timestep. A numerical residual can
only be judged to be large or small if the corresponding flows through the mesh
elements are known. To provide meaningful residual values, the CFX solver divides
by the appropriate scales at each point to normalize the residuals. The program
34
documentation recommends that “a reasonably converged solution requires a root
mean squared (RMS) residual level no higher than 5.0 x 10-5” [20]. The RMS
residuals were specified to run until variation of less than 1 x 10-10 from the previous
solution was calculated and double precision was used to obtain more accurate
solutions. Double precision eliminated roundoff errors at this tolerance level as seen
in a graph of the residuals in Figure 11. Using double precision, if the residual
tolerance was set low enough, roundoff errors would be seen at the machine precision
level of 1 x 10-14.
RMS P-Mass
1.E-01
RMS U-Mom
RMS V-Mom
RMS W-Mom
RMS residuals
1.E-03
1.E-05
1.E-07
1.E-09
1.E-11
1
21
41
61
Outer iterations
Figure 11: RMS residuals for two aligned 20μm diameter spheres at 10 m/s and a
separation of 2 diameters.
Roundoff errors manifest as residuals which fail to decrease with further
timesteps, instead of the linearly decreasing residuals which can be seen in Figure 11.
35
The RMS residual tolerance of 1 x 10-10 was reached before the specified 100 outer
iterations were exhausted, which means the number of outer iterations specified was
sufficient for convergence to this residual level. After the initial normal early timestep
oscillations, no further oscillations were observed in the residuals, indicating that
neither vortex shedding nor any other continuing transient behavior occurred and that
utilization of the laminar solver is appropriate.
Discretization errors between the converged simulation solution and the exact
solution cannot be assessed directly because the exact analytical solution is not known.
Running simulations with successively finer meshes will reduce the errors from the
mesh with each refinement until, with extremely fine meshes, roundoff error can
become dominant. Once the converged simulation solution in unchanging within a
close tolerance, the discretization errors from the mesh are sufficiently small to enable
adequately accurate numerical solutions in the absence of user and coding errors.
Grid convergence was studied using a single 20 μm diameter sphere having
0.6 μm, 1.2 μm and 2.4 μm grid spacings on the sphere surface and an inlet velocity of
10 m/s. The integrated force around the sphere, F, was output from the solver. The
drag coefficients, Cd, were then calculated using:
Cd 
2F
ρU 2 πa 2
(1)
with inlet velocity U, sphere radius a, and 1.185 kg/m3 for the air density, ρ, at 250C.
The drag coefficients obtained were 3.80, 3.65 and 3.62 for the 2.4 μm, 1.2 μm and
0.6 μm grids, respectively. The 2.4 μm and 1.2 μm grid Cd values only agreed to
within 5.2% and Cd values converged to within 0.8% for the 0.6 μm and 1.2 μm cases.
36
The grid convergence indicator (GCI) from Roache [23] indicates convergence with a
value of about 1. The 0.993 GCI calculated from these three Cd values demonstrate
that the solution has converged for the 0.6 μm grid refinement to an accuracy of
± 0.3%. The GCI calculation is shown in Appendix A. Drag coefficient values from
the 0.6 μm O-grid and the same O-grid with the addition of a refined mesh block
behind the sphere agreed to within 0.1%. The O-grid with a refined mesh block is
more similar to the two sphere models than the simple O-grid. Because the two sphere
models have the same meshed block, the O-grid with a block was used for all
validation simulations.
3.5 Validation
Validation of the one sphere solution was achieved by comparison to the
known accurate fit to experimental sphere drag data by White [2], given by:
Cd 
24
6

 0.4
Re 1  Re
(11)
Single 16 μm, 20 μm and 26 μm diameter spheres with air velocities of 6 m/s to
22 m/s were simulated in 2 m/s increments. Simulated Cd and Re values and the
percent deviations of the simulated values from the Oseen [3], Stokes [1] and White
[2] sphere drag equations, which are discussed in detail in Chapter 2 the Literature
Review and Test Plan section, are shown in Table 4. The Cd values from the models
agreee within ±1.8% with the values provided by White’s [3] experimental fit.
Figure 12 displays the simulated Cd values and drag coefficients from equations (1)
and (2) for Re < 40. Modeled Cd values deviated by 32% to 190% from the Oseen [3]
and Stokes [1] equations, but both the equations have large inaccuracies for Re > 1.
37
The close match to White’s [2] equation shown in Figure 12, validates that the
modeling implementation used accurately simulates sphere drag for the laminar
Re < 40 regime.
Table 4: Modeled sphere drag coefficients with errors from Oseen [3], Stokes [1] and
White [2].
Sphere
Diameter
(um)
Air
velocity
(m/s)
Re
Cd
CFX
Cd
Oseen
[3]
% from
Oseen
Cd
Stokes
[1]
% from
Stokes
Cd White
[2]
% from
White
16
16
16
20
20
20
26
26
26
26
26
6
10
14
6
10
14
6
10
14
18
22
6.2
10.3
14.5
7.7
12.9
18.1
10.1
16.8
23.5
30.2
36.9
6.07
4.21
3.35
5.17
3.62
2.90
4.30
3.05
2.47
2.12
1.88
8.38
6.83
6.16
7.60
6.36
5.83
6.88
5.93
5.52
5.29
5.15
-27.5%
-38.4%
-45.6%
-32.0%
-43.1%
-50.2%
-37.5%
-48.6%
-55.4%
-60.0%
-63.5%
3.88
2.33
1.66
3.10
1.86
1.33
2.38
1.43
1.02
0.79
0.65
56.6%
80.9%
101.7%
66.6%
94.6%
118.4%
80.3%
113.2%
141.2%
166.2%
189.1%
5.99
4.15
3.31
5.09
3.57
2.87
4.22
3.01
2.45
2.12
1.90
1.2%
1.4%
1.2%
1.6%
1.5%
1.1%
1.8%
1.4%
0.7%
-0.1%
-1.0%
10
Cd CFX
9
Cd Oseen [3]
8
Cd Stokes [1]
7
Cd White [2]
Cd
6
5
4
3
2
1
0
0
10
20
30
40
Re = Ud/ν
Figure 12: Modeled sphere drag coefficients with Oseen [3], Stokes [1] and White
[2] equations.
38
4 Data Reduction and Analysis
To generate results the drag forces must be determined and the drag
coefficients calculated from the data generated by the simulations. An iterative
method is used to calculate the transient trajectory behavior for single and multiple
spheres. Development of the required equations to determine these values is explained
in this section.
When a constant velocity field is imposed on a sphere a steady state pressure,
p, and shear stress, τ, distribution will develop around the sphere. The sphere can be
considered as a composition of very small surface elements given by dA. To
determine the total force on the sphere the pressure and shear forces at each element
on the surface of the sphere are integrated over the surface of the sphere. The integral
is given by:
F   p dA    dA
S
(30)
S
where S is the sphere surface. The drag force is given by the resultant force in the
direction of the approaching velocity field. For each differential element dA at an
angle theta from the direction of the flow field the drag force is given by [24]:
F   p dA sin     dA cos 
S
(31)
S
Both equations (30) and (31) will give the same values for these models, since there is
no net lift force. For the spheres considered here integration gives no net lift force
because the sphere and flow field are axisymmetric about an axis passing through the
39
spheres’ centers in the flow direction and thus the y and z force components cancel by
equal and opposite force components.
For each simulation the final converged values for the pressure and shear stress
force on the sphere are stored at each node, i, in the mesh that is on the sphere surface.
Each of these nodes has defined cartesian coordinates and an outward normal vector
which gives the orientation of the associated surface element, dA [20]. The total
pressure and shear stress force for each sphere is numerically integrated over the
surface of the sphere by the vector summation of the pressure and shear shear forces
for all nodes over the surface elements. The integrated pressure and shear stress forces
in the x-direction contribute to the drag force and are given by:
p x  i p xi dAi
(32)
 x  i  xi dAi
(33)
The vector components in each of the coordinates of the total pressure and shear stress
force are given in the solver output file for each sphere. The drag force on each sphere
is then obtained by adding the total pressure and shear stress force in the flow
direction x, given by:
F  px   x
(34)
Using the force values from equation (34) the drag coefficients for each sphere are
calculated from its definition, equation (1) , repeated here,
F 
C d ρU 2 As
2
(1)
40
with the frontal area, As, of πa2 for each sphere, the freestream velocity U and the
density of air, ρ, of 1.185 kg/m3 for atmospheric pressure air at 25OC.
For each sphere the Reynolds number, Re, is calculated using the diameter, d,
for that sphere, the freestream velocity U and the kinematic viscosity, ν = of 1.55 x10-5
m2/s for atmospheric pressure air at 25OC, as:
Re 
Ud
(35)

The nondimensional Reynolds number is used to compare the drag coefficients for
spheres with varying diameters and velocities.
Once drag coefficients are obtained the following equations and method are
used to calculate the transient response to simulate ink droplets. The volume for each
sphere is calculated from the diameter with the equation for the volume of a sphere
given by:
V
4a 3 3
[m ]
3
(36)
The density of the ink droplets is very close to that for water, so the mass for each
sphere is determined using the density for water, approximated to be ρ = 1000 kg/m3,
with the sphere volume by:
m  V 
 4a 3
3
[kg]
(37)
Newton's second law of motion was applied:
F  ma  m
dU
N 
dt
(38)
41
which is a first order ordinary differential equation (ODE) for the motion of a body in
a viscous fluid since F is a function of velocity, F(U). This can be rewritten as:
dU F U 

dt
m
(39)
Using a backward difference formula the derivative is given by the finite difference,
 dU  U n  U n1
 Ot 

 
t
 dt  n
(40)
where t is the timestep used in the calculation, n is the value at the current timestep
and n-1 is the value at the previous timestep [25]. This is a 1st order accurate
approximation with error on the order of t . With this backward difference the
velocity ODE can then be solved with the Explicit Euler method given by [25].
U n  U n -1 -
Fn -1 U t
m
(41)
In this form calculations can be solved explicitly as the current value depends only on
the previous values. For this method to yield a stable solution the timestep is limited
by: [23]
t 
2

where σ is the constants on the RHS of equation (39), which are discussed below.
Evaluating the drag force and mass for a sphere from equations (1) and (37),
respectively, gives:
(42)
42
 C d U ρU 2 πa 2 


2
2
F 
  3C d U U

m
2a
 4a 3  


 3 
(43)
Using the Cd equation (11) from White [2] results in a nonlinear equation. However,
using Stokes [1] Cd equation (8), the ODE from equation (39) conforms to the form of
the model problem, given by:
dU
 U
dt
(44)
so the stability requirement can be calculated [23]. Substituting Stoke’s [1] Cd,
equation (8), into equation (43) gives:
F 3U 2  24  18U


  2  U
m
2a  2aU 
a
(45)
which for a 20 μm diameter sphere yields a value for σ of 2.8 x 10-6 s-1. Using this
value for σ in equation (42) gives the maximum stable timestep of 7.2 x 10-7. To
assure stability a t of 5 x 10-7 was used for all calculations.
The explicit Euler method is also used to calculate the position of the ink
droplets, X, versus time with the equation:
Fn -1t 2
X n  X n-1  U n -1t 2m
(46)
For a single sphere, the trajectory calculation is initiated with a designated initial
velocity, Ui, equal to the approach velocity of the flow, and sphere diameter, and then
mass is then calculated from the diameter using equation (37). The initial distance, Xi,
is set to zero. The force on the sphere is calculated from equation (1) with the initial
43
velocity and the Cd value from White's [2] equation (11). For the 2nd timestep and
onwards the velocity is calculated using equation (41) and the position is calculated
from equation (46). At each timestep, the drag coefficient is calculated from White's
[2] equation (11) for the sphere velocity given by equation (41) at the current timestep
using:
Cdn 
24
6

 0.4
U n d 1  U n d /
(47)
The force is then calculated for the sphere velocity at the current timestep by
2
Cdn ρU n πa 2
Fn 
2
(48)
For the two sphere trajectory calculations, equations were developed which are
explained in Chapter 5, the Results and Discusiion section, to modify equation (47)
and incorporate the separation distance between the spheres. The solution procedure
is then the same as for the single sphere except that Equations (41), (46), (47) and (48)
are calculated separately for the leading and trailing spheres and the separation
distance and Reynolds numbers are updated at every timestep.
44
5 Results and Discussion
Presented are simulation results from the two sphere models for the test plan
conditions given in Tables 2 and 3. The single sphere model simulation results, which
were shown to match equation (11) given by White [2] within ± 1.8% are given in
Table 4. The Cd1, Cd2, Re1, and Re2 values calculated from the two sphere model
simulations are in Appendix B.
5.1 Drag Coefficient Evaluation
The calculated drag coefficients, Cd1 and Cd2, are evaluated versus δ, Re and U
to discern and quantify the trends and numerical relationships between the spheres.
With sufficiently large separation distance the spheres are expected to cease to
influence each other’s drag and obtain the same Cd as the equivalent sized single
sphere.
5.1.1 Equal Sized Spheres
For clarity the drag behavior of a single geometry, the 20 μm diameter equal
sized spheres, is presented first. For both the leading and trailing spheres, a separate
Cd curve is provided across the range of separation diameters, δ, for each applied
freestream velocity, as shown in Figure 13.
6
6
5
5
4
4
Cd2
Cd1
45
3
2
3
2
6 m/s
8 m/s
10 m/s
12 m/s
14 m/s
1
6 m/s
8 m/s
10 m/s
12 m/s
14 m/s
1
0
0
1
2
3
δ
(a)
4
5
6
1
2
3
δ
4
5
6
(b)
Figure 13: Drag coefficients for two aligned 20μm diameter spheres (a) leading (b)
trailing as a function of separation distance, δ.
The Cd values are lower and have a larger change in magnitude for both the leading
and trailing spheres as the separation distance is reduced. There is a large variation,
on the order of two in Cd1 versus Cd2 for each δ value, over the range of velocities.
Consequently, Cd does not collapse into single curves for the leading and trailing
spheres. Following the typical presentation of single sphere Cd values from Figure 12,
Cd1 and Cd2 are plotted in Figure 14 versus Reynolds number, Re, for the modeled
range of separation distances. The range of studied velocities are thus collapsed into
the Reynolds number.
6
6
5
5
4
4
Cd2
Cd1
46
3
δ =1.5
δ=2
2
δ=3
δ=4
2
1
δ=5
1
3
δ =1.5
δ=2
δ=3
δ=4
δ=5
White[2]
White[2]
0
0
0
5
10
Re1
(a)
15
20
25
0
5
10
Re2
15
20
25
(b)
Figure 14: Drag coefficients for two aligned 20μm diameter spheres versus Reynolds
number with single sphere predictions from White [2]: (a) leading (b) trailing.
However, for each δ there is a separate set of Cd versus Re values. At each δ the Cd
values describe a curve of similar shape to the single sphere curve, as seen for δ = 1.5
and δ = 5 in Figures 15a and 15b, respectively. This figure also shows that the Cd
values increase for both the leading and trailing spheres with greater separation
distances. At δ = 5 the leading sphere value of Cd versus Re approaches the Cd
relationship for a single sphere, while the trailing sphere Cd is still at least 20% lower.
This trend demonstrates that the leading sphere recovery to the single sphere Cd value
occurs at closer distances than the recovery for the trailing sphere. To express the
drag coefficients as a function of Re, separate curve fits could be made to the unique
Cd curves for each δ, but the mathematics would quickly become unruly. Specifically,
the δ values scale more tightly with δ at high Re and more broadly with δ at low Re, as
47
shown in Figures 15a and 15b, so obtaining a general expression for in-between δ
6
6
5
5
4
4
Cd
Cd
values would be difficult.
3
2
3
2
Cd1
1
Cd1
1
Cd2
Cd2
White [2]
White [2]
0
0
0
5
10
15
Re
(a)
20
0
5
10
15
20
Re
(b)
Figure 15: Drag coefficients for two aligned 20μm diameter spheres versus Re for two
separate distances (a) δ = 1.5 (b) δ = 5.
A more convenient way to express the drag for two spheres with various
separation distances is by using a drag reduction parameter, λ. Each drag coefficient
value, Cd1 or Cd2, is divided by the Cd for a single sphere using equation (11) from
White [2] to give the drag reduction parameter λ. For the leading and trailing spheres,
respectively, the λ definitions are:
1 
Cd 1
Cd
(49)
2 
Cd 2
Cd
(50)
48
The λ1 and λ2 values for 6 m/s to 14 m/s velocities in 2 m/s increments are given
versus separation distances δ in Figure 16. Note that the scale for λ1 and λ2 starts at
1.0
1.0
0.9
0.9
0.8
0.8
λ2
λ1
0.4.
0.7
0.6
0.7
0.6
6 m/s
8 m/s
10 m/s
12 m/s
14 m/s
0.5
6 m/s
8 m/s
10 m/s
12 m/s
14 m/s
0.5
0.4
0.4
1
2
3
δ
(a)
4
5
6
1
2
3
δ
4
5
6
(b)
Figure 16: Drag reduction parameter as a function of separation distance for two 20μm
spheres (a) leading (b) trailing.
In Figure 16a the drag reductions approaches zero (i.e. λ1 approaches 1) at large
separation distances and are significantly higher at close distances. By δ = 5 the
leading sphere reaches λ1 = 0.98, or a Cd almost equivalent to a single sphere. Drag
reduction increases at closer distances to a maximum of about 15% for λ1 near where
the spheres just touch at a δ of 1. The λ1 values for the leading sphere are largely
coincident for all velocities. Insensitivity to velocity is termed Re independence.
Single spheres exhibit Re independence which enables their drag coefficients to be
described by the single curve given by White [2], which is shown in Figure 12. The
leading sphere drag coefficients, therefore, could be fairly well described by a single
49
curve. The largest spread in λ1 values for the velocities studied is 1.7% for the δ = 1.5
distance and decreases with δ to the smallest spread of 0.3% for δ = 5. The λ1 values
are slightly higher for larger applied velocities at all values of δ.
Drag reduction for the trailing sphere is more significant, resulting in drag
coefficients ranging from about 50% to 80% of those for a single sphere at the
separation distances and velocities simulated. For the trailing sphere, the λ2 values
show a wider spread for the range of applied velocities than was evident in the λ1
values. A maximum of 10.5% spread in λ2 values is observed at δ = 1.5 and contracts
somewhat with δ to a 6.3% spread at δ = 5. Values for λ2 tend to approximately
decrease by 0.006 to 0.008 for every 1 m/s increase of velocity.
The λ values calculated from Stimson’s [7] equation (3) in Table 1 are shown
in Figure 17 with the leading sphere and trailing sphere values at 10 m/s along with
the average value of λ for both spheres given by:
avg 
1  2 
2
(51)
50
1.0
0.9
0.8
λ
0.7
0.6
0.5
λ1
λ2
0.4
λavg
Stimson [7]
0.3
1
6
11
16
21
δ
Figure 17: Drag reduction parameters for two aligned 20μm spheres at 10 m/s: leading
and trailing.
These simulations which are carried out to higher δ values than shown previously,
show that the influence of the leading sphere on the trailing sphere persists beyond 20
diameter separations, as the λ2 values have only reached 0.94 by 19 diameters.
Beyond this point λ increases very slowly. In contrast, the leading sphere has
recovered to 99% of the single sphere Cd by 6 diameters.
Stimson’s [7] equation (3) predicts the same value of λ for each of the two
aligned equal sized spheres in a very slow creeping flow where inertia is neglected.
However, these λ values agree very well with λave values for these higher Re flows
with substantial inertia effects. As seen in Figure 12 for Re > 12 the drag coefficients
from White’s [2] empirical equation (11) for a single sphere are approximately double
those predicted by Stokes’s [1] solution equation (8), which neglects inertia, indicating
51
that for the Re = 6 to 24 range modeled in this study inertia effects are important and
ultimately influence pressure and stress fields. The close correlation between
Stimson’s [7] solution and the simulated λavg values might suggest that the total
combined drag reduction is solely a function of the pressure and viscous effects
resulting from the flow geometry and independent of inertia effects. However, the
inertia associated with the approach velocity is a means to affect the distribution of the
local drag reduction between the leading and trailing spheres. The influence due to
inertia on drag reduction is quantified as the deviation of the leading (or trailing)
sphere λ from the λavg for two sphere systems, as shown in Figure 18.
0.35
0.30
0.25
λ
0.20
0.15
λ1- λavg
1 - λavg
0.10
0.05
0.00
1
6
11
16
21
δ
Figure 18: Inertia and geometric contributions to drag reduction parameter λ.
52
For a single sphere λ = 1, therefore, the influence on drag reduction due to the two
sphere configuration is given by 1- λavg, and is a consequence of a lesser reduction for
the leading sphere and greater reduction for the trailing sphere λ. The value of λavg for
these simulations are within ± 3% of that given by Stimson [7] for the 6 m/s to 14 m/s
range of flow velocities, as shown in Figure 19 for the two ends of this range.
0.9
0.9
0.8
0.8
0.7
0.7
λ
1.0
λ
1.0
0.6
0.6
0.5
λ1
λ2
0.5
λ1
λ2
0.4
λavg
0.4
λavg
Stimson [7]
Stimson [7]
0.3
0.3
1
2
3
δ
(a)
4
5
1
2
3
4
5
δ
(b)
Figure 19: Drag reduction parameter for two aligned 20μm diameter spheres as a
function of δ and velocities of (a) 6 m/s (b) 14 m/s.
5.1.2 Unequal Sized Spheres
The drag reduction associated with unequal sized spheres was studied since
they closely represent observed ink droplets, which generally have a leading droplet
larger than the trailing droplets. Drag reduction parameters for a 26 μm leading sphere
with an equal sized, or smaller, trailing sphere at the velocity of 10 m/s are shown in
Figure 20.
1.0
1.0
0.9
0.9
0.8
0.8
λ2
λ1
53
0.7
0.7
0.6
0.6
β = 0.62
β = 0.62
β = 0.77
0.5
β = 0.77
0.5
β=1
β=1
0.4
0.4
1
2
3
(a)
δ
4
5
6
1
2
3
δ
4
5
6
(b)
Figure 20: Drag reduction parameter for two aligned spheres, d1 = 26 μm, at 10 m/s
and as a function of δ for different size ratios β: (a) leading (b) trailing.
Smaller sized trailing spheres, i.e. β < 1, have lower λ2 values compared with equal
size spheres, and thus larger drag reductions. The greatest spread in λ2 values for the
sphere sizes studied is 0.1 at δ = 1.5. Conversely, a smaller sized trailing sphere
results in less drag reduction on the leading sphere. The 0.08 maximum spread in λ1
over these β values is also less than the spread for the trailing sphere. Combining the
λ1 and λ2 values for unequal size sphere pairs results in average drag reduction
parameter values that are similar to those for equal sized spheres, as shown in
Figure 21.
54
1.0
0.9
0.8
λ
0.7
0.6
0.5
λ1
λ2
0.4
λavg
Stimson [7]
0.3
1
2
3
4
5
δ
Figure 21: Drag reduction parameters for two aligned spheres, d1 = 26 μm, at 10 m/s as
a function of and for β of 0.62, 0.77, and 1.
Interestingly, even though the Stimson [7] equation (3) was developed for equal sized
spheres the simulation results for λavg for unequal sized spheres at 10 m/s are within
± 3%. As separation distances increase, unequal sized spheres also have a decreasing
effect on the drag reduction. Variation in λ due to the sphere size ratio β drops below
2% for both leading and trailing spheres at distances greater than 5 separation
diameters.
5.1.3 All Spheres
The wide range of possible drag reduction parameters is shown in Figure 22,
for sphere separations of δ = 1.5 to 5 for all β and U values studied from Table 2.
Note that estimates for λ from Stimson [7] are also provided for δ = 1 and δ = 1.13.
The general trend of decreasing drag reduction with increasing δ is shown.
55
1.0
0.9
0.8
λ
0.7
0.6
0.5
λ1
λ2
0.4
λavg
Stimson [7]
0.3
1
2
3
4
5
δ
Figure 22: Drag reduction parameter versus sphere spacing for all test cases in Table
2. For λ2 values of λ are lower with smaller β, higher U and unchanged by d1. For λ1
values of λ are lower with higher β, and unchanged by U and d1.
The drag reduction parameters at δ = 1.5, which have the highest variation, span from
0.87 to 0.92 and from 0.37 to 0.56 for the leading and trailing spheres, respectively,
with a 0.66 to 0.71 spread for the average drag reduction.
5.2 Surface Force Distributions
The total drag force is a combination of both shear stress and pressure forces
acting on the surface of the spheres. At every point the pressure force, p, is the force
component normal to, and the shear stress, τ, is the force component tangential to, the
surface of the sphere. The velocity slope at the sphere surface is given by the shear
strain from the shear stress, τ, and air viscosity μ.
u 

[1/s]
r 
(52)
The pressure and shear strain distributions around a single sphere are reported in [24]
56
and [26] and develop as follows. The highest pressure is at the stagnation point on the
center of the frontside of the sphere. At this point the total force from the flow is
normal to the sphere. There is no force tangential to the sphere surface because the
flow bifurcates and the net shear stress is zero. As the fluid moves around the sphere
the pressure force decreases and shear stress increases because the freestream velocity
becomes more tangential to the sphere surface. Once the shear stress reaches a
maximum the pressure becomes less than that in the freestream as the fluid transitions
from compressive on the sphere surface to expansive from the sphere surface.
5.2.1 Closely Separated Spheres
The shear strain and pressure distributions around the surfaces of the spheres
give insight to the sources of the drag reduction with separation distance and relative
sphere sizes. All the leading spheres in the figures below have a 26 μm diameter.
Note that the reference pressure is 30 Pa so pressures below this are essentially
negative pressures as they are lower than the freestream pressure. The most
significant deviation from the single sphere force distributions is at the lowest value of
δ. Pressure and shear strain distributions around the leading sphere at δ = 1.5 are
shown for a 6 m/s flow in Figure 23.
57
80
18
Single sphere
70
Shear Strain Rate [ 10 /s ]
β = 0.77
60
Pressure [ Pa ]
β = 0.62
50
40
30
20
0
10
135
β = 0.77
14
β=1
5
β=1
0
180
Single sphere
16
β = 0.62
12
10
8
6
4
2
90
45
0
180
0
135
90
45
0
θ
θ
(a)
(b)
Figure 23: Leading sphere surface distributions at 6 m/s for d1 = 26μm and δ = 1.5
(a) pressure (b) shear strain.
Distributions around the trailing sphere for the same flow are given in Figure 24.
80
18
Single sphere
70
Shear Strain Rate [ 10 /s ]
β = 0.77
60
Pressure [ Pa ]
5
β=1
50
40
30
20
10
0
180
Single sphere
16
β = 0.62
β = 0.62
β = 0.77
14
β=1
12
10
8
6
4
2
135
90
θ
(a)
45
0
0
180
135
90
45
0
θ
(b)
Figure 24: Trailing sphere surface pressure and shear strain distributions at 6 m/s for
d1 = 26μm and δ = 1.5 (a) pressure (b) shear strain.
58
Figure 25 shows the corresponding single sphere pressure and shear strain
distributions for the 16 μm, 20 μm and 26 μm single spheres for comparison to the
trailing sphere distributions at the end of this section. Note that these sphere sizes
correspond to β = 0.62, 0.77, and 1, respectively.
80
18
d = 16
70
Shear Strain Rate [ 10 /s ]
Pressure [ Pa ]
d = 20
d = 26
60
14
d = 26
trailing β = 1
5
trailing β = 1
50
40
30
20
10
0
180
d = 16
16
d = 20
12
10
8
6
4
2
135
90
θ
(a)
45
0
0
180
135
90
45
0
θ
(b)
Figure 25: Single sphere surface pressure and shear strain distributions at 6 m/s and
δ = 1.5 (a) pressure (b) shear strain.
For the leading sphere in Figure 23 there is a pressure recovery on the trailing
side when it is followed closely by the trailing sphere such as the case δ = 1.5. The
maximum pressure recovery at the rear stagnation point, shown at zero degrees on the
figures, is 51%, 36% and 26% for β = 1, 0.77 and 0.62 trailing spheres, respectively.
There is also shear strain reduction where the shear strain drops to near zero for a
region on the backside of the sphere for δ = 1.5. The near zero shear strain region
extends about ± 30 degrees from the rear stagnation point for all three β sizes studied.
As observed in Figure 24 the near zero shear strain region extends for a similar
59
distance from the front stagnation point on the trailing spheres. Also the pressure
distribution on the frontside of the trailing sphere, is decreased to values close to that
of the freestream pressure 30 Pa. Pressures near the rear of the leading sphere drop
slightly below freestream values, as is noted in Figure 23a. These pressure and shear
strain distribution similarities indicate that the flow conditions at the rear of the
leading sphere and front of the trailing sphere are very similar. Inspection of the
velocity contours in Figure 26 shows that there is a large lower velocity region
spanning between the spheres that may be contributing to the rather low pressure
deviations and low strain rates observed in Figures 23 and 24.
(a)
(b)
Figure 26: Velocity contours for 6 m/s, d1 = 26 μm, δ = 1.5 (a) β = 0.62 (b) β = 1.
The shear strain and pressure contours given in Figures 27 and 28,
respectively, show a near zero shear strain thoughout this low velocity region and
pressure near that of the freestream around the rear stagnation point. Further
inspection of the low velocity region with vector plots given in Figure 29 shows that a
60
recirculating wake has developed and that the velocity is very low, near 0.1 m/s
throughout most of the wake.
(a)
(b)
Figure 27: Shear strain contours for 6 m/s, d1 = 26 μm, δ = 1.5 (a) β = 0.62 (b) β = 1.
(a)
(b)
Figure 28: Pressure contours at 6 m/s, d1 = 26 μm, δ = 1.5 (a) β = 0.62 (b) β = 1.
61
(a)
(b)
Figure 29: Velocity vectors for 6 m/s, d1 = 26 μm, δ = 1.5 and β = 1 (a) both spheres
(b) zoomed in view.
The trailing spheres have a very similar low velocity region near their front stagnation
point and, consequently, very similar shear strain and pressure distributions.
Recall, Figure 24, which shows that the trailing sphere pressure and shear
strain distributions are very similar for all three β cases. However, the λ2 values,
provided in Table 5, are quite different with values of 0.55, 0.49 and 0.46 for β = 1,
0.77 and 0.62, respectively. Figure 25 shows that the single spheres, with diameters of
16 μm, 20 μm and 26 μm, used to calculate λ2 from equation (50), have higher
pressure and shear strain peaks for smaller diameter spheres, which correspond to
lower β values. Thus, because higher peaks indicate higher Cd values, which are in the
denominator of equation (50), lower β trailing spheres will have lower λ2 values.
For a single sphere situation, the integrated shear stress based on the second term in
equation (31) and the integrated pressure on the first term in equation (31) were
62
assessed. The average ratio of shear stress to pressure forces based on all the
simulations is about 2:1. Using this ratio and the fraction of the peak pressure and
shear strain rates for the trailing sphere β = 1 case, in Figure 25, versus those for the
same size single sphere, also from Figure 25, results in an estimated λ2 of 0.58 which
is a reasonable estimate of the simulated λ2 value of 0.52. This illustrates that
comparison of the peak pressure and shear strain ratios is a useful way to estimate the
total drag reduction.
5.2.1 Spheres with Multi-diameter Separation
As the separation distance between spheres increases the velocity field changes
significantly. Consequently the pressure and shear strain at the sphere surface are also
affected. At δ = 5 separation the pressure and shear strain distributions around the
leading sphere are almost indistinguishable between the various sphere pairs and a
single sphere, as shown in Figure 30.
80
18
Single sphere
70
Pressure [ Pa ]
Shear Strain Rate [ 10 /s ]
β = 0.77
60
β = 0.62
β = 0.77
14
β=1
5
β=1
50
40
30
20
10
0
180
Single sphere
16
β = 0.62
12
10
8
6
4
2
135
90
θ
(a)
45
0
0
180
135
90
45
θ
(b)
Figure 30: Leading sphere surface distributions at 6 m/s for d1 = 26μm and δ = 5
(a) pressure (b) shear strain.
0
63
These distributions are in agreement with the calculated drag reduction, λ1, of less than
2% for this separation distance for all sphere size combinations. Trailing sphere
distributions for a leading sphere of 26 μm and trailing spheres corresponding to
β = 0.61, 0.77 and 1.0 are shown in Figure 31. Unlike the case for the leading sphere,
trailing sphere pressure and shear strain distributions are still influenced at this δ = 5
spacing. The d1 = 26 μm, β = 1 pressure and shear strain distributions are reproduced
along with single sphere distributions for all three sphere sizes in Figure 32. Again for
the β = 1 case taking the ratio of the maximum pressure and maximum shear strain for
the trailing sphere over the single sphere values and the average 2:1 viscous to
pressure force ratio from the simulations gives a λ2 estimate of 0.79 compared to the
actual simulated λ2 value of 0.78.
18
80
Single sphere
70
Shear Strain Rate [ 10 /s ]
Pressure [ Pa ]
β = 0.62
β = 0.77
60
14
β = 0.77
β=1
5
β=1
50
40
30
20
10
0
180
Single sphere
16
β = 0.62
12
10
8
6
4
2
135
90
θ
(a)
45
0
0
180
135
90
45
θ
(b)
Figure 31: Trailing sphere surface pressure and shear strain distributions at 6 m/s,
δ = 5 (a) pressure (b) shear strain.
0
64
18
80
d = 16
70
Shear Strain Rate [ 10 /s ]
Pressure [ Pa ]
d = 20
d = 26
60
14
d = 26
trailing β = 1
5
trailing β = 1
50
40
30
20
10
0
180
d = 16
16
d = 20
12
10
8
6
4
2
135
90
θ
(a)
45
0
0
180
135
90
45
0
θ
(b)
Figure 32: Single sphere surface pressure and shear strain distributions at 6 m/s and
δ = 5 (a) pressure (b) shear strain.
The trailing sphere distributions at δ = 5 have no noticeable shape differences
compared to the single spheres, the peak values are just lower. From this similarity to
the single sphere distributions, perhaps the trailing sphere has drag behavior close to
that of a single sphere, but with a reduced, albeit not uniform, “freestream” velocity
which develops behind the leading sphere. The velocity contours in Figure 33 show
that a flow varying from 3 m/s to 5 m/s is present about midway between the 26 μm
spheres. Because λ2 is 78% of the single sphere value, if the drag reduction were due
solely to the reduced velocity field, the average velocity in front of the trailing sphere
would be 0.78 times 6 which is  to 4.7 m/s. This is within the range of velocities
shown by the contours. The pressure and shear strain contours in Figure 34 show that,
unlike for the δ = 1.5 case, both spheres have similarly shaped distributions with little
interaction between the spheres, at least within the resolution shown in the figures.
65
The flattened front of the trailing sphere shear strain distribution compared against the
rounded front of the leading sphere distribution shows that the velocity onto the
trailing sphere is uneven and lower towards the front stagnation point.
Figure 33: Velocity contours for 6 m/s, d1 = 26 μm, δ = 5, β = 1
(a)
(b)
Figure 34: Contours for 6 m/s, d1 = 26 μm, δ = 5, β = 1 (a) pressure (b) shear strain
66
The λavg drag reduction for all equal sized spheres simulated is shown in Figure
35 with ½ the λ value for an ellipsoid in a prolate orientation from Clift’s [17]
equation (19). The total length for the ellipsoids is δ + 1 to be comparable with the
δ + 1 distance from the front of the leading sphere to the rear of the trailing sphere. At
δ = 3 an ellipsoid and sphere pair have about the same drag reduction relative to each
other. At closer distances the ellipsoid has larger drag reduction than a sphere pair,
likely due to its more aerodynamic shape compared to the sphere pair. At distances
above δ = 3 the ellipsoid has less drag reduction indicating that the shear stress from
its greater surface area has increasingly more of an effect.
1.2
λavg
1.0
0.8
0.6
λavg Spheres
1/2 Ellipsoid
0.4
1
3
5
δ
7
9
Figure 35: Drag reduction parameter versus sphere spacing with ellipsoid predictions
from Clift[17].
67
5.3 Drag Reduction Characterization
To describe the drag behavior of two spheres under all conditions studied,
equations for Cd1 and Cd2 as a function of the separation distance and sphere velocities
are developed. The equations presented use the drag reduction parameters, λ1 and λ2,
which are then multiplied by the single sphere Cd from the White [2] equation (11) to
give the Cd1 and Cd2 values. Separate equations for the leading and trailing spheres are
developed and the equations are specified for a given range of separation distances, δ.
5.3.1 Unaffected Distance
With increasing separation distance the velocity and pressure fields defining
the wake behind the leading sphere should recover to that for a single sphere, and Cd1
should approach the value for a single sphere. From Figure 16, the unaffected
distance where Cd1 is equivalent to the single sphere Cd to within 1% occurs at δ = 6.
Consequently for δ greater than 6 the single sphere White [2] equation (11) is used for
Cd1, or:
Cd1 = Cd for δ > 6
(53)
Similarly, the trailing sphere at a sufficiently large δ should reach a point at
which there is no influence from the leading sphere. Within the modeled domain of
these simulations, the trailing sphere never achieves a Cd2 value equivalent to a single
sphere. However, by extrapolation of a power series curve fit to the λ2 values
between δ = 11 and 19, it is found that λ2 is approximately 0.99 at 32 diameters. The
drag coefficients are then determined by the power series fit for λ2 multiplied by the
White [2] equation (11) drag coefficients. The result is:
68
Cd 2  Cd 2  C d 0.68 0.11
for 11    32
(54)
The power series equation has a standard error of the fit for the linearized equation
given by:
for 11    32
Y  0.38  2.21X  0.002
(55)
Power series equation (54) is shown in Figure 36. The methodology to determine the
standard error of the fit is given in Appendix C.
1.0
0.9
λ2
0.8
0.7
0.6
0.5
0.4
10
12
14
16
18
20
δ
Figure 36: Trailing sphere drag reduction versus δ for equal sized spheres with
equation (54) for δ  11
For δ  32 the spheres are assumed to no longer interact and Cd2 is assumed to equal
the single sphere value:
Cd 2  Cd
For δ > 32
(56)
The main focus of this modeling is for   5 where drag reduction effects are
strongest and most likely to cause appreciable drag difference with different sphere
configuration. Equations (54) and (56) are developed using only 20 μm diameter
69
equal sized sphere pairs at a 10 m/s velocity. Because this velocity is the midpoint of
the simulated range and because drag differences due to velocity and β lessen with
separation distance, this simplification is thought to provide adequate estimation of the
drag reduction.
5.3.2 Trailing Sphere Equations
Recall that for the trailing sphere there is considerable spread in the λ2 values
for each separation distance, δ, due to the variation in velocity as shown in Figure 16b
and size ratio β as shown in Figure 20b. Based on just the equal sized sphere cases,
i.e. d1 = d2, λ2 values are found to vary linearly with Re10.5 for each separation distance.
The linear equations for δ = 1.5 and 5 are shown in Figures 37a and 37b, respectively.
1.0
1.0
δ=5
0.9
0.9
0.8
0.8
λ2
λ2
δ = 1.5
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
2.0
2.5
3.0
3.5
Re1
(a)
0.5
4.0
4.5
5.0
2.0
2.5
3.0
3.5
Re1
4.0
4.5
0.5
(b)
Figure 37: Trailing sphere drag reduction versus Re10.5 for equal sized spheres for
(a) δ = 1.5 (b) δ = 5
It is found that the individual slopes for fits at δ = 1.5, 2, 3, 4 and 5 vary from the
average slope of -0.036 by ± 8%. The curve fit equations for each δ value were
5.0
70
modified to account for the intercepts, I2 as follows. Substitute -0.036 for the slope
coefficients from each fit, and then reevaluate the associated intercepts by replacing
the λ2 values with:
I 2  2  (0.036 Re10.5 )
for δ<11
(57)
Expressing the intercepts, I2, as a function of δ-1, which is the distance between
trailing and leading stagnation points of the leading and trailing spheres, respectively,
results in a single curve fit with variation less than 2% for each δ. Intercept values and
equation (58) are shown in Figure 38.
I 2  0.725  1
0.148
(58)
1.1
1.0
I2
0.9
0.8
0.7
0.6
0.5
0
2
4
6
δ-1
8
10
12
Figure 38: Equal sized trailing spheres values of I2 versus intersphere distance δ-1.
Using equation (58) for unequal sized sphere pairs results in I2 values up to 17%
below the fit value at the closest separation distance of δ = 1.5, as seen in Figure 39.
71
1.1
1.0
I2
0.9
0.8
0.7
β=1
0.6
β = 0.62, 0.77, 0.8
0.5
0
2
4
6
8
10
12
δ-1
Figure 39: Trailing sphere intercept values, I2, versus distance (δ-1) for different
sphere size ratios β.
The highest values of I2 at each δ are for the equal sized spheres, and the I2 values
become lower as β decreases. A factor, Φ2, was sought to increase the I2 values as a
function of β so that results for all trailing spheres would collapse onto a single curve.
For the approach velocity of 10 m/s, d1 = 26 μm and β = 0.62, 0.77 and 1 the factor
values were calculated from:
2 
2  
2   1
(59)
for δ = 1.5 to 5, since for this d1 there are three β cases which will enable a check on
the curvature of the relationship to be developed. The Φ2 values are related to the
separation distances using linear fits versus 1/(δ-1), as shown in Figure 40.
72
1
0.98
0.96
0.94
Φ2
0.92
0.9
0.88
β = 0.62
0.86
β = 0.77
0.84
Linear ( β = 0.62 )
0.82
Linear ( β = 0.77 )
β=1
Linear ( β = 1)
0.8
0
0.5
1
1.5
2
2.5
1/(δ-1)
Figure 40: Trailing sphere Φ2 values versus 1/(δ-1) with associated linear curve fits.
The resultant equations are:
2 
 0.082
 0.994
  1
for   0.62
(60)
2 
 0.051
 0.996
  1
for   0.77
(61)
for  1
(62)
2  1
At separation distances greater than δ = 5, or 1/ (δ-1) < 0.2, which are in the upper left
corner of Figure 40, λ2 values for the different sized trailing spheres agree with λ2
values for equal sized spheres within about 2%. Thus, it was judged that the use of a
Φ2 factor to adjust for sphere size differences at greater distances is unnecessary.
Using Φ2 versus 1/(δ-1) forces all the linear equation intercepts to be 1 because the
drag reduction parameter, λ2, converges to 1 for all β values as 1/(δ -1) goes to zero at
large separation distances. Rather a linear equation was found for the slopes of Φ2
73
versus 1/(δ-1) from (60), (61) and (62) and reported as a function of the sphere size
ratio β in Figure 41.
0
0
0.2
0.4
0.6
0.8
1
1.2
-0.01
-0.02
-0.03
M
-0.04
-0.05
-0.06
-0.07
-0.08
-0.09
β
Figure 41: Slope, M, of I2 versus 1/(δ-1) for a range of β.
The slopes were found to scale linearly to the sphere size ratio β such that:
M  0.215  .0.215
(63)
Replacing the slopes in equations (60) and (61) with the right hand side of equation
(63) enables Φ2 to be expressed as a single equation in terms of the known β and δ
values, given by:
2 
0.215  .0.215  1
  1
The intercepts, I2, are divided by Φ2 from equation (64) and plotted versus δ - 1 in
Figure 42.
(64)
74
1.1
1.0
I2/Φ2
0.9
0.8
0.7
0.6
0.5
0
2
4
6
8
δ-1
Figure 42: Trailing sphere intercept values, I2, with β adjustment factor, Φ2, versus
distance (δ-1) for different sphere size ratios β.
Employing a power series fit, yields a standard error of the fit of
Y  0.33  1.90 X  0.19
for 11    32
(65)
for the linearized equation. The maximum deviation of modeled values from the fit is
reduced to less than 2% for most cases compared with 17% without using Φ2. The
exception is the β = 0.8 case which has modeled values that deviate up to 5% from the
fit.
I2
0.15
 0.72  1
2
(66)
Rearranging equation (66) for I2 substituting into equation (57) solving for λ2 and
multiplying by the single sphere drag yields:

Cd 2  Cd 2  C d  2 0.72  1
0.15
 0.036 Re10.5

for 1    11
(67)
75
The drag coefficient values for the trailing sphere, Cd2, can now be calculated at all
separation distances, 1    11, 11    32 and δ >32 using equations (54), (56)
and (67), respectively.
5.3.3 Leading sphere equations
Arriving at equations for the leading sphere drag is simplified compared to
those of the trailing sphere because, as shown in Figure 16a, there is little velocity
dependence on λ1 values for the leading sphere. The power series fit of λ1 for equal
sized spheres versus δ - 1 is shown in Figure 43 and yields λ1 values at each δ that are
within 2% of experimental values reported for each velocity.
1  0.915  10.054 for  1
(68)
1.1
1.0
λ1
0.9
0.8
0.7
0.6
0.5
0
1
2
3
4
5
6
δ-1
Figure 43: Leading sphere drag reduction 1 for equal sized spheres versus intersphere
distance (δ - 1).
76
Once unequal sized sphere pairs are added to the dataset as shown in Figure 44,
experimental λ1 values can deviate up to 8% from those predicted from equation (68).
Deviations are higher for lower separation distances.
1.1
1.0
λ1
0.9
0.8
0.7
β=1
β = 0.62, 0.77, 0.8
Power ( β = 1)
0.6
0.5
0
1
2
3
4
5
6
δ-1
Figure 44: Leading sphere drag reduction, λ1, for all sized spheres versus intersphere
distance (δ-1).
Following the same process as for the trailing sphere, a Φ1 factor is given by:
1 
0.094  0.094   1
  1
(69)
The power series fit of λ1 for all sized spheres versus (δ-1), including the Φ1
modification, is then:
1
1
 0.92  1
0.05
(70)
which has an standard error of the fit of:
Y  0.08  2.99 X  0.01
(71)
77
for the linearized equation, and data points with a maximum deviation of 2% from the
fit, as shown in Figure 45.
1.1
1.0
λ1/Φ1
0.9
0.8
0.7
0.6
0.5
0
1
2
3
4
5
6
δ-1
Figure 45: Leading sphere drag reduction normalized by Φ1 for all sized spheres
versus intersphere distance (δ-1).
The equation for Cd1 for the lower δ range where C d1 is not equivalent to a single
sphere is given by:

Cd1  Cd 1  C d 1 0.92  1
0.05

for 1    6
(72)
5.4 Transient Drag Calculation
The velocity of real droplets in a viscous fluid is continually retarded by the
drag forces. To determine realistic trajectories for both single spheres and sphere pairs
the equations of motion equations (41) and (46) are used to incorporate the changes in
droplet velocity and drag force as the spheres decelerate. The transient drag
calculations and implementation for a single sphere are explained in detail in the Data
78
Reduction and Analysis chapter, Chapter 4. Modifications to these equations for two
aligned spheres are explained in the following paragraph.
The initial specified parameters for two spheres are the diameter and velocity
for each of the two spheres and the center-to-center distance, δ, between the spheres.
The initial position for the trailing sphere is set at zero and the position for the leading
sphere is set at the separation distance. For each timestep the velocity and position for
each sphere are calculated. The separation distance is recalculated after each timestep
by subtracting the relative positions of the two spheres for that same timestep. The
drag coefficients are determined for each timestep from equations (53) or (72) for the
leading sphere and from equations (54), (56) or (67) for the trailing sphere, depending
on the separation distance at that timestep. The output from the calculation is the
distance traveled and velocity of each sphere and relative separation distance between
the two spheres for each timestep up to a specified time or until the time when the
spheres collide. For equal sized 20 μm diameter water droplets with a starting
separation of δ = 4, both with an initial velocity of 10 m/s, a collision time of 190 μs
is predicted as shown when the intersphere distance goes to zero in Figure 46.
79
0.0020
3.5
0.0018
3.0
0.0016
x [m]
0.0012
2.0
0.0010
1.5
0.0008
0.0006
δ-(d 1+d2)/2
2.5
0.0014
1.0
0.0004
Leading sphere
0.5
Trailing sphere
0.0002
δ-(d1+d2)/2
0.0000
0.0
50.0
100.0
150.0
0.0
200.0
time (μs)
Figure 46: Predicted model output d1 = 20, β = 1, both with initial velocities of 10 m/s,
and δ = 4.
To characterize the design tool developed, times to collision, tc, and distances to
collision, xc, were computed for cases in the test plan at separation distances of δ = 5
and less. The distances to collision for the equal sized spheres in Figure 47 show that
the larger spheres travel further before colliding. Spheres at higher velocity, as
observed in Figure 47b, also travel further before colliding. The relationship of xc to δ
appears to be linear in both Figures 47a and 47b. Note that the collision distance for
all size spheres is given by:
  d avg   
d1  d 2
2
(73)
80
0.0035
0.0035
0.003
0.003
0.0025
0.0025
0.002
0.002
6 m/s
8 m/s
xc [m]
xc [m]
10 m/s
0.0015
12 m/s
14 m/s
0.0015
0.001
0.001
d = 20 μm
d = 26 μm
0.0005
0.0005
0
0
0
1
2
3
4
5
6
0
1
2
3
(a)
4
5
δ
δ
(b)
Figure 47: Equal sized spheres collision distance xc versus δ for both d1 and all U (a)
d1 labeled (b) U labeled
Referring back to equations (1) and (36) combined and repeated here as equation (74)
 C d ρU 2 πa 2

2
dU F 
 
dt
m
 4a 3  


 3 


2
  3C d U
2a
(74)
Temporal changes in velocity are proportional to the ratio of force over mass. Sphere
velocity is expected to decay more slowly, for larger sphere sizes and/or lower
velocities, because F/m is smaller for these cases. Figure 47 shows that for a typical
2 mm distance between an inkjet cartridge and the paper, equal sized spheres will
collide when separated by no more than 3 diameters for the 14 m/s, 26 μm diameter
case and up to 5 diameters for the 6 m/s, 20 μm diameter case.
Dividing the collision distance by the leading sphere diameter, d1, gives a nondimensional collision distance:
6
81
c 
xc
d1
(75)
Near linear relationships of ηc as a function of Re, with different lines for each
separation distance, are shown in Figure 48.
0.00014
0.00012
0.0001
ηc
0.00008
0.00006
δ = 1.5
δ =2
0.00004
δ =3
δ =4
0.00002
δ =5
0
0
5
10
15
20
25
Re
Figure 48: Equal sized spheres non-dimensional collision distance ηc to Re
Expressing ηc per separation diameter collapses the data fairly well into a single line as
shown in Figure 49.
82
0.00003
0.000025
ηc /δ
0.00002
0.000015
δ = 1.5
0.00001
δ =2
δ =3
0.000005
δ =4
δ =5
0
0
5
10
15
20
25
Re
Figure 49: Equal sized spheres non-dimensional collision distance ηc by δ to Re
The non-dimensionalized collision distance is then given by:
 c  2.1 x 10 -5 
 20%
for 7 < Re < 24
(76)
Because there is some curvature in the data a more accurate estimate of the nondimensionalized collision distance can be obtained with a power series fit to the data
for δ of 5 or less given by:
 c  1x10 5  0.23
for 7 < Re < 24
(77)
which has a standard error of the fit for the linearized equation of:
Y  11.5  1.5 X  0.4
for 7 < Re < 24
(78)
All equal sized spheres will collide given enough time. The collision time, tc ,
is near linear with separation distance for δ = 5 and lower, as shown in Figure 50.
83
0.0006
0.0006
0.0005
0.0005
6 m/s
8 m/s
10 m/s
12 m/s
0.0004
0.0004
tc [s]
tc [s]
14 m/s
0.0003
0.0002
0.0003
0.0002
d = 20 μm
0.0001
0.0001
d = 26 μm
0
0
0
1
2
3
4
5
6
0
1
2
3
δ
(a)
4
5
δ
(b)
Figure 50: Equal sized spheres collision time tc versus δ for both d1 and all U (a) d1
labeled (b) U labeled
For the cases tested the collision time is more dependent on the sphere velocity
whereas the collision distance is more dependent on the sphere mass, which is shown
by Figures 47 and 50. Collision time can be non-dimensionalized using velocity and
leading sphere diameter by:
c 
t cU
d1
(79)
Similarly as for the collision distance, the non-dimensional collision time, τc , has a
unique line for each separation diameter, δ, as shown in Figure 51.
6
84
0.00014
0.00012
0.0001
τc
0.00008
0.00006
δ = 1.5
δ =2
0.00004
δ =3
δ =4
0.00002
δ =5
0
0
5
10
15
20
25
Re
Figure 51: Equal sized spheres non-dimensional collision time, τc , to Re
The values are much more tightly distributed for τc on a per δ basis as shown in
Figure 52.
0.00003
0.000025
τc /δ
0.00002
0.000015
δ = 1.5
0.00001
δ =2
δ =3
0.000005
δ =4
δ =5
0
0
5
10
15
20
25
Re
Figure 52: Equal sized spheres non-dimensional collision time, τc , by δ to Re
85
With the exclusion of the δ = 1.5 data, the non-dimensionalized collision time can be
expressed as:
 c  2.4 x 10 -5 
 15%
for 7 < Re < 24
(80)
and a more accurate estimate of the non-dimensionalized collision time can be
obtained with a power series fit to the data given by:
 c  1x10 5  0.18
for 7 < Re < 24
(81)
which has a standard error of the fit for the linearized equation of:
Y  11.5  1.5 X  0.7
for 7 < Re < 24
(82)
Using equations (75), (76), (79) and (80) an estimate of sphere collision time and
distance can be made for specific starting conditions.
For unequal sized sphere pairs, the decreased drag on the smaller trailing
sphere is counteracted by faster loss of momentum compared to the leading sphere.
Using the term from equation (74) leads to the following condition, which needs to be
satisfied for collision of unequal sized spheres:
C d 2U 22 C d 1U 12

a2
a1
(83)
Basically, the trailing sphere needs to decelerate more slowly than the leading sphere
for a collision to occur. Rewriting equation (83) in terms of the sphere size ratio
gives:
U
Cd 2
   1
Cd1
U2
2

  

(84)
86
where the right hand side of the equation reduces to β when the sphere velocities are
the same. This equation shows that when there is more drag improvement from the
separation distance, than there is relatively less deceleration from the differences in
mass, the spheres will collide.
In a convenient form this collision criteria is given by:

Cd 2
0
Cd1
(85)
The trajectories for a δ that just converges and a δ that just diverges, along with the
collision criteria from equation (85) are shown in Figures 53 and 54, respectively, for
the β = 0.8 case at 10 m/s initial velocity.
0.0070
2.0
0.25
1.8
0.20
0.0060
1.6
0.15
1.2
0.0040
1.0
0.0030
0.8
β-Cd2/Cd1
1.4
δ-(d 1+d2)/2
x [m]
0.0050
0.10
0.05
0.00
0.6
0.0020
-0.05
Leading sphere
0.0010
Trailing sphere
δ-(d1+d2)/2
0.0000
0.0
200.0
400.0
600.0
800.0
0.4
0.2
0.0
1000.0 1200.0
-0.10
-0.15
0.0
0.5
1.0
1.5
time (μs)
δ
(a)
(b)
2.0
2.5
Figure 53: β = 0.8 spheres at 10 m/s just converging with δ = 2.695 (a) trajectory (b)
collision criteria
3.0
87
0.0070
2.5
0.25
0.20
0.0060
2.0
0.15
0.0030
1.0
β-Cd2/Cd1
1.5
0.0040
δ-(d 1+d2)/2
x [m]
0.0050
0.10
0.05
0.00
0.0020
-0.05
Leading sphere
0.0010
0.5
-0.10
Trailing sphere
δ-(d1+d2)/2
0.0000
0.0
200.0
400.0
600.0
800.0
0.0
1000.0 1200.0
-0.15
0.0
0.5
1.0
1.5
2.0
time (μs)
δ
(a)
(b)
2.5
3.0
3.5
Figure 54: β = 0.8 spheres at 10 m/s just diverging with δ = 2.700 (a) trajectory (b)
collision criteria.
None of the unequal sized sphere cases converge for δ greater than 2, therefore
presentation of data as a function of δ are too sparse to be meaningful. Instead the
maximum δ distance where a collision will still occur, δc, was computed for each case
and shown in Figure 55.
88
3.5
3
2.5
δc
2
1.5
1
β = 0.62
β = 0.77
0.5
β = 0.80
0
0
5
10
15
20
25
Re
Figure 55: Maximum distance for collision, δc, for smaller trailing sphere versus Re
Figures 53 and 54 show that the δc for the β = 0.8 spheres at 10 m/s case is 2.695,
because at the slightly higher δ of 2.700 the spheres diverge. There is a limit close to
the β = 0.62 case where a trailing sphere will never converge with the leading sphere
even if the separation distance is very close. From equation (85) this limit is
equivalent to the ratio of the drag coefficients for the leading and trailing spheres,
which is also equivalent to the ratio of the drag reduction parameters. Therefore, from
the ratio of the drag reduction at any δ the maximum β sphere that will collide can be
predicted.
89
6 Conclusion
In this study, CFD simulations of the flow around two rigid aligned spheres
using atmospheric pressure air applied at constant velocity were used to generate
non-dimensional correlations of steady state drag coefficients. Models were created
for all test cases in the testplan and initiated with constant velocity air at 6 m/s, 10 m/s
and 14 m/s. Models were created with the following sphere size combinations: equal
sized 20 μm and 26 μm diameter sphere pairs, a 20 μm leading sphere with a 16 μm
trailing sphere, and a 26 μm leading sphere with a 16 μm trailing sphere as well as a
20 μm trailing sphere. In the models the spheres were positioned with center-to-center
separation distances, δ, of 1.5 to 19 sphere diameters. The flows generated were in
the Reynolds number range of 5 to 25.
At these modeled conditions significant drag reduction was found to occur,
especially for the trailing sphere. For equal sized spheres, the leading sphere drag
reduction compared to that of a single sphere was found to be independent of Re,
“within 1.7%”. For δ < 5 drag reduction increases for closer distances to a maximum
of about 15% when the spheres just touch, at δ = 1. The trailing sphere has an average
drag reduction of 20% at δ = 5, which increases to 50% at δ = 1.5. There are spreads
of up to 10.5% in the drag reduction for the range of air velocities studied, with larger
drag reduction occurring at the higher velocities. Larger size spheres in the same flow
have lower drag coefficients due to the increased Reynolds number, but there appears
to be no effect on the drag reduction with sphere size. Averages of the leading sphere
90
and trailing sphere drag reduction for each case are within ± 3 % of the creeping flow
solution values given by Stimson [7].
For unequal sized spheres drag reduction on the smaller trailing sphere
increases, and on the leading sphere lessens, compared to the equal sized spheres. The
drag reduction difference compared with equal sized spheres is near zero at δ = 5, but
does increase with the closer spacings. For the most dissimilar size sphere pair at the
closest distance of δ = 1.5 a further 9% drag reduction for the trailing sphere, for a
total drag reduction compared to a single sphere of 54%, is predicted compared to the
same velocity and equal sized sphere. For this sphere pair drag reduction of 6% less
on the leading sphere is also predicted, for a total drag reduction compared to a single
sphere of 6%.
Evaluation of the pressure and shear strain distributions at the surface of the
spheres showed that the primary contributor to the drag reduction on the trailing
sphere at all separation distances, is the reduced velocity field created in front of the
trailing sphere. For the range of Reynolds numbers studied, the reduced velocity field
causes drag reduction on the trailing sphere at separations up to 32 diameters. The
leading sphere has drag reduction caused by modification to its wake region, with
pressures closer to freestream pressures in the neighborhood of the rear stagnation
point, which reduces the pressure and shear stress forces compared to a single sphere.
At the closest separation of δ = 1.5 there is additional drag reduction on both the
leading and trailing spheres from a large, very low velocity region between the
spheres. The flow recirculates in this region and the near zero velocity results in near
91
zero values of both the pressure deviations from the atmospheric pressure and shear
strain on the surface of the spheres in the areas that are in contact with this
recirculation region.
From the drag reduction parameters for all the sphere size cases, empirical
equations were found for the drag coefficients as a function of separation distance.
These equations were then used in a backward Euler iterative calculation to estimate
the sphere trajectories and the collision time and distance for two spheres based on the
initial diameters, velocity and separation distance. Although equal sized spheres were
found to always collide given sufficient time, to result in a collision of droplets within
the 2 mm distance typical of the distance between an inkjet cartridge and the paper,
the droplets would need to have an initial separation of no more than 3 to 5 diameters.
Furthermore, for inkjet applications, to enable droplets to collide, the ratio of the
trailing sphere to leading sphere diameters must be greater than 0.62 and preferably at
least 0.75. Otherwise the increased drag reduction on the trailing sphere is
compensated for by the faster loss of momentum due to the smaller mass. Even with
diameter ratios greater than 0.75, droplets will tend to diverge unless they are initially
separated by only 2 to 3 diameters.
92
Bibliography
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study of the drag on a sphere at intermediate Reynolds and Peclet numbers. J.
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a sphere at low and moderate Reynolds numbers. J. Fluid Mech. 48, pp 771789.
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treating multiparticle slow viscous flow: axisymmetric flow past spheres and
spheroids. J. Fluid Mech., v50, p4, pp705-740.
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Academic Press, NewYork, pp 99-106.
14. Hadamard, J. 1911. Mouvement permanent lent d’une sphere liquide visqueuse
dans un liquide visqueux. C. R. Acad. Sci. Paris Ser. A-B, v152, pp 17351739.
93
15. Rybczynski, W., 1911. Uber die fortschreitende Bewegung einer flussigen
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40-46.
16. Beard, K.V. and Pruppacher, H.R. 1969. J. Atmos. Sci. 26, pp 1066-1072.
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18. Michaelides, E.E. 1973. Review – the transient equation of motion for
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19. Rowe, P.N. 1961. The drag coefficient of a sphere, Trans. Inst. Chem. Engr.
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20. 2009. Ansys Release 12.1 Documentation
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22. Rhie, C.M. and Chow, W.L. 1983. Numerical study of the turbulent flow past
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94
Appendices
95
Appendix A – GCI Calculation from Roache [23]
Grid size
0.6 μm
1.2 μm
2.4 μm
Grid space ratio r
1
2
4
Cd
3.62
3.65
3.80
 3.80  3.65 
ln 

3.65  3.62 
p 
 2.32
ln 2
GCI12 
GCI24 
FS 12
r
3.62  3.65
3.62
 0.00259 x100%  0.26%
2.32
2 1
1.25

1

FS  24
1.25
r
p

(A1)

3.65  3.80
3.65
 0.0129 x100%  1.29%
2.32
2 1

(A3)
GCI24
 1.29

 0.993  1 so indicates convergence
p
r GCI12 4.997(0.26)
(A4)
p
1

(A2)
96
Appendix B – Table 5: Calculated two sphere drag from simulations
U [m/s] d1 [μm] d2 [μm]
6
6
6
6
6
10
10
10
10
10
14
14
14
14
14
6
6
6
6
6
8
8
8
8
8
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
β
Cd1
Cd2
Re1
Re2
λ1
λ2
λavg
δ
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
4.71
4.84
4.94
5.08
5.12
3.34
3.42
3.48
3.57
3.60
2.70
2.75
2.80
2.87
2.89
4.51
4.70
4.94
5.04
5.10
3.71
3.86
4.04
4.12
4.15
3.21
3.21
3.32
3.41
3.47
3.52
3.55
3.58
3.60
3.60
3.61
3.61
3.62
3.62
3.62
3.62
3.63
3.63
3.63
3.64
3.64
3.65
3.13
3.55
3.87
4.48
4.73
2.00
2.31
2.53
2.97
3.16
1.50
1.74
1.93
2.29
2.44
2.89
3.18
3.58
3.86
4.06
2.28
2.52
2.85
3.08
3.25
1.91
1.91
2.11
2.27
2.40
2.51
2.60
2.75
2.86
2.94
3.01
3.07
3.13
3.18
3.22
3.25
3.27
3.30
3.32
3.34
3.36
3.38
7.74
7.74
7.74
7.74
7.74
12.90
12.90
12.90
12.90
12.90
18.06
18.06
18.06
18.06
18.06
7.74
7.74
7.74
7.74
7.74
10.32
10.32
10.32
10.32
10.32
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
6.19
6.19
6.19
6.19
6.19
10.32
10.32
10.32
10.32
10.32
14.45
14.45
14.45
14.45
14.45
7.74
7.74
7.74
7.74
7.74
10.32
10.32
10.32
10.32
10.32
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
12.90
0.91
0.93
0.95
0.98
0.99
0.92
0.94
0.96
0.98
0.99
0.92
0.94
0.95
0.98
0.99
0.87
0.92
0.95
0.97
0.98
0.88
0.91
0.95
0.97
0.98
0.88
0.88
0.91
0.94
0.95
0.97
0.97
0.98
0.99
0.99
0.99
0.99
0.99
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.52
0.59
0.64
0.74
0.78
0.48
0.55
0.60
0.71
0.75
0.45
0.52
0.58
0.68
0.73
0.56
0.62
0.70
0.75
0.79
0.54
0.60
0.68
0.74
0.78
0.53
0.53
0.59
0.63
0.67
0.70
0.72
0.76
0.79
0.82
0.84
0.85
0.87
0.88
0.89
0.90
0.91
0.92
0.92
0.93
0.93
0.94
0.71
0.76
0.80
0.86
0.88
0.70
0.74
0.78
0.85
0.87
0.68
0.73
0.77
0.83
0.86
0.72
0.77
0.82
0.86
0.89
0.71
0.76
0.82
0.86
0.88
0.71
0.71
0.75
0.78
0.81
0.83
0.85
0.87
0.89
0.90
0.91
0.92
0.93
0.94
0.94
0.95
0.95
0.96
0.96
0.96
0.97
0.97
1.5
2
3
4
5
1.5
2
3
4
5
1.5
2
3
4
5
1.5
2
3
4
5
1.5
2
3
4
5
1.5
1.5
2
2.5
3
3.5
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
97
Appendix B (continued) – Table 5: Calculated two sphere drag from simulations
U [m/s] d1 [μm] d2 [μm]
12
12
12
12
12
14
14
14
14
14
16
16
16
16
16
6
6
6
6
6
10
10
10
10
10
14
14
14
14
14
6
6
6
6
6
10
10
10
10
10
14
14
14
14
14
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
β
Cd1
Cd2
Re1
Re2
λ1
λ2
λavg
δ
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.62
0.62
0.62
0.62
0.62
0.62
0.62
0.62
0.62
0.62
0.62
0.62
0.62
0.62
0.62
0.77
0.77
0.77
0.77
0.77
0.77
0.77
0.77
0.77
0.77
0.77
0.77
0.77
0.77
0.77
2.85
2.95
3.08
3.14
3.18
2.59
2.67
2.79
2.85
2.87
2.39
2.46
2.56
2.62
2.64
4.07
4.14
4.22
4.26
4.28
2.91
2.95
3.00
3.03
3.04
2.36
2.39
2.43
2.45
2.46
3.97
4.07
4.19
4.24
4.27
2.84
2.90
2.98
3.02
3.04
2.31
2.35
2.41
2.44
2.46
1.65
1.83
2.09
2.27
2.40
1.46
1.63
1.86
2.03
2.15
1.32
1.47
1.69
1.84
1.95
2.77
3.29
3.96
4.35
4.64
1.72
2.09
2.57
2.86
3.07
1.25
1.55
1.94
2.18
2.36
2.53
2.91
3.43
3.74
3.97
1.62
1.89
2.27
2.49
2.66
1.20
1.43
1.74
1.92
2.07
15.48
15.48
15.48
15.48
15.48
18.06
18.06
18.06
18.06
18.06
20.65
20.65
20.65
20.65
20.65
10.06
10.06
10.06
10.06
10.06
16.77
16.77
16.77
16.77
16.77
23.48
23.48
23.48
23.48
23.48
10.06
10.06
10.06
10.06
10.06
16.77
16.77
16.77
16.77
16.77
23.48
23.48
23.48
23.48
23.48
15.48
15.48
15.48
15.48
15.48
18.06
18.06
18.06
18.06
18.06
20.65
20.65
20.65
20.65
20.65
6.19
6.19
6.19
6.19
6.19
10.32
10.32
10.32
10.32
10.32
14.45
14.45
14.45
14.45
14.45
7.74
7.74
7.74
7.74
7.74
12.90
12.90
12.90
12.90
12.90
18.06
18.06
18.06
18.06
18.06
0.88
0.91
0.95
0.97
0.98
0.88
0.91
0.95
0.97
0.98
0.88
0.91
0.95
0.97
0.98
0.94
0.96
0.98
0.99
0.99
0.95
0.96
0.98
0.99
0.99
0.95
0.96
0.97
0.98
0.99
0.92
0.94
0.97
0.99
0.99
0.93
0.95
0.97
0.98
0.99
0.92
0.94
0.96
0.98
0.98
0.52
0.57
0.65
0.71
0.75
0.50
0.56
0.64
0.70
0.74
0.49
0.55
0.63
0.69
0.73
0.46
0.54
0.65
0.72
0.77
0.41
0.50
0.61
0.68
0.73
0.37
0.46
0.58
0.65
0.70
0.49
0.57
0.67
0.73
0.77
0.45
0.53
0.63
0.69
0.74
0.42
0.49
0.60
0.66
0.71
0.70
0.74
0.80
0.84
0.87
0.69
0.74
0.80
0.84
0.86
0.69
0.73
0.79
0.83
0.86
0.70
0.75
0.82
0.85
0.88
0.68
0.73
0.80
0.83
0.86
0.66
0.71
0.78
0.82
0.84
0.71
0.76
0.82
0.86
0.88
0.69
0.74
0.80
0.84
0.86
0.67
0.72
0.78
0.82
0.85
1.5
2
3
4
5
1.5
2
3
4
5
1.5
2
3
4
5
1.5
2
3
4
5
1.5
2
3
4
5
1.5
2
3
4
5
1.5
2
3
4
5
1.5
2
3
4
5
1.5
2
3
4
5
98
Appendix B (continued) – Table 5: Calculated two sphere drag from simulations
U [m/s] d1 [μm] d2 [μm]
6
6
6
6
6
10
10
10
10
10
14
14
14
14
14
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
β
Cd1
Cd2
Re1
Re2
λ1
λ2
λavg
δ
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
3.79
3.94
4.12
4.21
4.25
2.72
2.81
2.93
2.99
3.02
2.21
2.28
2.37
2.42
2.44
2.34
2.58
2.94
3.16
3.34
1.55
1.73
1.99
2.15
2.28
1.20
1.34
1.55
1.69
1.79
10.06
10.06
10.06
10.06
10.06
16.77
16.77
16.77
16.77
16.77
23.48
23.48
23.48
23.48
23.48
10.06
10.06
10.06
10.06
10.06
16.77
16.77
16.77
16.77
16.77
23.48
23.48
23.48
23.48
23.48
0.88
0.91
0.96
0.98
0.99
0.89
0.92
0.96
0.98
0.98
0.88
0.91
0.95
0.97
0.98
0.55
0.61
0.69
0.74
0.78
0.51
0.57
0.65
0.71
0.75
0.48
0.54
0.63
0.68
0.73
0.71
0.76
0.82
0.86
0.88
0.70
0.74
0.81
0.84
0.87
0.68
0.73
0.79
0.83
0.85
1.5
2
3
4
5
1.5
2
3
4
5
1.5
2
3
4
5
99
Appendix C – Standard error of the fit calculation for power series
Using the least squares curve fitting functions in the Excel ® computer
program the data were found to be related by exponential regression fits of the form:
  b c
(A5)
The error in each curve fitting function was assessed with its precision interval [27].
To determine the precision interval the equation is linearized as:
ln   ln b  c ln 
(A6)
which can be rewritten in the standard linear form as:
Y = B + mX
(A7)
The curve fit with its precision interval is given by:
95%
Y  t v,95 S yx
(A8)
for these cases where the variance in the curve fit line is assumed to be not due to the
independent X values. The Student t distribution, t v ,95 , is specified at a 95%
confidence interval for the degrees of freedom, v. For a regression fit of polynomial
order m to N data points the degrees of freedom, v, is given by:
v = N- (m+1)
(A9)
For linear fits the order of m is 1. A polynomial describes the data to a precision given
by the standard error of the fit:
N
S yx 
 y
i 1
 y ci 
2
i
v
(A10)
100
where yi are the data values and yci are the predicted values from the regression fit.
Calculations of the precision intervals for the linearized forms of the exponential
regression fits are reported in the Results and Discussion section, Chapter 5.