Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
A Reinterpretation of the Odar and Hamilton Data on the Unsteady Equation of Motion of
Particles
Efstathios E. Michaelides and Adam V. Roig
Department of Mechanical Engineering
The University of Texas at San Antonio
San Antonio TX, 78249, USA
stathis.michaelides@tutsa.edu, eom302 t@my.utsa.edu
Keywords: history term, added mass term, unsteady motion, equation of motion
Abstract
Since their publication in 1964 the experimental data and correlations derived by Odar and Hamilton have been the basis of
several studies that included the history (Basset) term in the expression of the unsteady drag on particles. However, recent
studies have shown that the value of the added mass coefficient is constant, 1/2, over a very large range of Reynolds numbers.
Also, recent analytical studies on the history term have proven that the form of the history term used by Odar and Hamilton is
not correct for high Reynolds numbers (the history term decays faster when fluid advection is significant). However, the
experimental data are accurate at very low Reynolds numbers and, most probably, they represent the most reliable set of
experimental data on the measured total unsteady force on solid spheres. Based on the manifest accuracy of the total
unsteady force that emanates from the Odar and Hamilton correlation and the fact that the added mass term and the
quasisteady term have been accurately determined from other analytical or experimental studies, we conducted a study to
recalculate the functional form of the history term in the unsteady equation of motion at low Reynolds numbers and to derive
a new correlation for the socalled "history force coefficient," AH. The new correlation applies to low Reynolds numbers and
is expressed in terms of the Reynolds and Stokes numbers of the particle. This short paper offers details on the computations
of this new correlation for the history term as well as a quantitative analysis on the importance of this term within the range of
Stokes numbers of the original data.
Introduction
Several practical applications ranging from coal and droplet
combustion in boilers and burners to the transport of
aerosols in the atmosphere involve the unsteady motion of
solid and fluid particles. Boussinesq (1885) and Basset
(1888) first derived the unsteady equation of motion of a
small spherical solid particle at Re<
conditions), which in terms of the drag force exerted by the
fluid on the particle, Fi, may be written as follows:
d(u, v)
4F 3 d(u, v, )+R2p dt' dt(1)
F, = 6 Ru(u, v, )+/ R 3l .. .. + R ("p/) f1 '
3 ) dt *u rtt'
where Fi is the total force acting on the particle; vi and ui
are the velocities of the particle and the fluid respectively; p
the density of the fluid; t is the time; R is the radius of the
sphere; and i is the viscosity of the fluid. The first term in
Eq. (1) represents the friction drag on the particle. The
second term is called "added mass" or "virtual mass" and
accounts for the fluid that is accelerated or decelerated with
the particle. The last term represents the cumulative effect
of the unsteady motion of the particle on the fluid velocity
field and is called the "history term" or the "Basset term."
Equation (1) is often times referred to as the
"BoussinesqBasset expression."
Because a great deal of the practical applications of
gassolid flows occur at higher Reynolds numbers, several
researchers tried to extend this expression to higher
Reynolds numbers. Expressions for the steady drag term
at higher Re were derived from experimental data, as for
example, the expression by Schiller and Nauman (1933),
which is still used today in steadystate applications.
Regarding the transient terms of the unsteady equation of
motion for hydrodynamic force, Odar and Hamilton (1964)
conducted an extensive set of experiments and were the first
to propose empirical coefficients for all the terms of Eq. (1)
which would make the equation valid at higher Re.
Thus, Odar and Hamilton proposed the following form for
the unsteady drag force acting on a spherical particle:
F =C 21( vv)2 +C, 3 d(uv,)
2 3 idt
d(u, v,)
dt' dt' (2)
fto Ir7
where CD, CA and CH are three empirical coefficients,
which were given by correlation functions, derived from
experimental data.
Of the three coefficients, CD is the steadystate drag
coefficient. Odar and Hamilton (1964) used the Schiller
and Nauman (1933) correlation for it:
Paper No
CD [1+ 0.15 Re0687 (3)
Re
where the Reynolds number is defined as:
2p\u, v, R
Re = (4)
The other functions, CA and CH, were obtained from the
experimental data of Odar and Hamilton (1964). The
correlations proposed are as follows:
0.066
C, = 1.05 (5)
Ac2 +0.12
and
CH =2.88
3.12
+ 1)3
(Ac +1
The acceleration number, Ac, was defined as follows:
AC (i i )2
Ac=
2R d(u, v) (7)
dt
The empirical equation and the coefficient correlations were
verified later by Odar (1966) who used data of the total
unsteady drag force exerted by the fluid. Since their
publication, the expressions for the unsteady hydrodynamic
force on particles have been used in numerous analytical
and computational studies. An extensive treatment on the
transient equation of motion at creeping flow conditions as
well as at finite Reynolds numbers, which includes several
studies that have verified the Odar and Hamilton
correlations for particulate flow, may be found in the review
article by Michaelides (2003) and the monograph by
Michaelides (2006).
THE NEED TO REINTERPRET THE ODAR AND
HAMILTON DATA
Regardless of the results of analytical studies that focused
on the decays of the transient terms, the general practice of
using the Odar and Hamilton equation, that is Eq. (2), to
describe the unsteady motion of spherical particles was
deemed successful in the past and has been verified by
several other studies as for example, by Tsuji et al. (1991).
The calculations emanating from this expression showed
good agreement with sets of experimental data. However, all
these studies used and verified the total hydrodynamic force
and not the three separate parts of it. Thus, the use of the
Odar and Hamilton expression became popular in
engineering calculations since the 1970's because it yielded
accurate results. The reason for this manifested accuracy
and the popularity of this transient expression is the relative
ease of calculations with the aid of computers and the close
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
agreement of the results with experimental data. The
agreement with experimental data is rather fortuitous and
due to the following reasons:
a) The equation has a sound experimental basis,
since it is a correlation of accurate experimental results, and,
b) The equation has been derived from
measurements of the entire hydrodynamic force and has
always been used in calculations to determine the entire
hydrodynamic force, and not any of its three parts,
separately.
It must be noted that the experiments, from which the
coefficients of the Odar and Hamilton expression as well as
other semiempirical expressions emerged, measured the
total hydrodynamic force on the sphere under various flow
conditions. Then, by a series of assumptions and deductions,
the experimentalists estimated the three parts of the
expressions and, hence, determined the three coefficients,
CD, CA and CH, as functions of parameters, such as Re and
Ac. Similarly, for any verification of the derived expressions
in other studies that followed, the total hydrodynamic force
was always computed as a single entity. Any comparisons
were done for the entire hydrodynamic force as determined
by computations with sets of experimental data, which were
similar to the ones from which the force was determined in
the first place. Therefore, even if the calculations of the
three parts of the hydrodynamic force, separately, were
laden with high errors, the calculation of their sum, which is
equal to the total hydrodynamic force, would have had very
low error. Because of this coincidence in the
"determination" and "verification" of the results, it would
have been rather surprising if close agreement between
"experiments" and "theory" were not obtained. This
fortuitous agreement is the main reason why semiempirical
expressions, such as the one by Odar and Hamilton, are still
considered accurate enough and are being used extensively
in transient flow calculations. The derived results for the
hydrodynamic force, as a whole, can be trusted within their
range of applicability because they have a sound
experimental basis.
Regarding the three separate terms in Eq. (2), several
analytical and experimental results have shown that the
added mass coefficient, CA, in Eq. (2) is constant and equal
to '2. The origin of this term is from potential flow theory
(Poison, 1831) and there is no reason why the terms should
be a function of the Reynolds number or the acceleration
number, Ac. Several analytical studies in the past
confirmed that CA is constant. In addition, Bataille et al.
(1990) used the experimental technique of Darwin (1953)
and conducted a series of experiments, which showed
unequivocally that the added mass coefficient is equal to /2
in the range 0
These considerations for the coefficient CA, cast significant
doubts that the two last terms of Eq. (2) accurately represent
the added mass and the history term in the unsteady
equation of motion. The coefficient of the first term, CD,
which represents the steady drag on a spherical particle was
determined by independent experimental steady drag data
and is considered to be accurate. However, Odar and
Hamilton (1964) measured the total transient hydrodynamic
force in a series of accurate experiments. Then they
subtracted the steady drag using the drag coefficient as
expressed in Eq. (3) and, calculated the sum of the history
and added mass terms. Subsequently, using a series of
Paper No
inferences Odar and Hamilton (1964) separated the effect of
the other two terms, and derived separate correlations for CA
and CH. Therefore, the sum of the last two terms in Eq. (2)
stems from accurate experimental data, even though the
inferences that gave rise to the correlations (5) and (6) may
not be correct. If CA=1/2 for a wide range of Reynolds
numbers, then correlation (6) must be reevaluated to yield
an accurate expression for the history term.
The experimental study by Odar and Hamilton (1964) is
considered to be the most detailed experimental study on the
transient equation of motion or spherical particles. Despite
the error in the separate determination of the added mass
and the history terms, the experimental data themselves for
the total hydrodynamic force, F,, are accurate. Therefore, it
is good to use this set of accurate and meticulously obtained
data in order to obtain a more accurate representation of the
history term. For this reason, we undertook to reinterpret
the experimental data by Odar and Hamilton (1964) under
the light of the more recent studies and the known, constant
coefficient for the added mass term. From this
reinterpretation of the data a new correlation was derived
for the coefficient, CH, of the history term.
ANALYSIS OF THE EXPERIMENTAL DATA
At first, Eq. (2) was cast in a form that is more commonly
used by investigators in recent years as follows:
d(u ) (8)
FC p(u,vIA,) A dp +,ARR2(pp)7' d dpt'
2 2 3 dt t'
According to independent experimental data, and accurate
expression for CD is given by Eq. (3), while an accurate
expression for AA is AA=1 (or CA=1/2) for the entire range of
the acceleration and Reynolds numbers of the original data
by Odar and Hamilton. This allowed us to use the entire
set of the original data in order to determine the functional
form of the parameter AH in Eq. (8), or CH in Eq. (2).
The original set of experimental data of the Odar and
Hamilton (1964) study is not available. For this reason, we
followed the original study and "backtracked" through the
expressions Odar and Hamilton used to calculate their final
correlation. The original study used a solid spherical
particle with oscillatory motion in a fluid at rest to measure
the total transient hydrodynamic force. We followed the
same expressions Odar and Hamilton used to reduce the
original set of data and performed several numerical in uk'
to compute the total hydrodynamic force on the oscillating
sphere with a set of similar frequencies and motion
amplitudes as in the original study. The parameters used in
each trial are shown in Table A, below.
Amplitude
Radius, of motion, Frequency, maximum Stokes maximum
Trial # m m 1/s Re number Ac
1 005 02 03 007 000751 28
2 005 1 1 1 13 002504 2260
3 01 1 1 225 010015 1130
4 002 01 02 009 000080 36
5 001 1 05 012 000050 920
6 025 05 2 563 125194 226
7 002 008 1 0472 0038 000420 61E6
8 0 025 01 01 0006 000063 69
9 004 05 08 036 001282 214
of the numerical
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The values of the fluid properties used to reconstruct the
original experimental data in the nine trials were the same as
in the original study: p=889.07 kg/m3 and g= 79.002
Ns/m2, respectively. The Stokes number, St, which appears
on Table A, is a dimensionless expression of the
characteristic time of the particle divided by the
characteristic time of the oscillatory motion. In this case,
the angular frequency of the oscillations, co, was used for the
computation of the Stokes numbers:
8R2.p, 167rJR2p
St = )Tp ,= C
9p, 9,Uf
where f is the frequency of oscillation of the particle and pp
the material density of the particle. Rather low frequencies
of the order of 1 s1 were used in the original study by Odar
and Hamilton and similar frequencies were used in the trials
of the present study. The Stokes number is the parameter
that is more commonly used in the oscillatory type of flow,
such as the one in the original study.
Following the method Odar and Hamilton used to calculate
their results, we backtracked their equations and computed
the total instantaneous hydrodynamic force for 28 time
instances in each trial set. Thus, we computed the total
hydrodynamic force, F1, on the oscillating particle for 252
time values, which is approximately equal to the number of
data points in the original set of experimental data. Using
the same expression for CD as in the original study and the
more accurate expression AA=1 (or CA= 12) a new value for
the history force, FHNEW, was computed as follows:
R' =1 c, p(v, 1 ) a m, d(v, u')+ 1 d(v, u) d(v, u,)
F F Cr1? pv u) A A ,n 05.rlp 1
2 2 dt L 2 dt 3 dt
(10)
Hence, the parameter, AHNEW was calculated for every
time instance in the nine "trials" by the following
expression:
AH
tiNE
FH
d(v, u)
tdt'
All the calculations were performed by Microsoft Excel in a
personal computer.
ANALYSIS OF THE COMPUTED DATA
Figure 1 shows the instantaneous values of the parameter
AH, as it was calculated from the original Odar and
Hamilton correlation and as it was computed from Eq. (11).
The results pertain to the conditions identified as trials 1
and 2 in Table A and are plotted as a function of the
instantaneous Reynolds number. It is observed that
there is a significant difference in the data calculated by
the two expressions, especially at the lower values of the
Reynolds number. It is also observed that the points
computed by the revised expression, Eq. (11) show less
scatter. Such observations lead to the conclusion that the
Reynolds number is a better parameter to express the
functional relationship of the history terms than the
Table A: Particle and flow parameters
trials
Paper No
originally used acceleration number, Ac.
6Ea on T6 3l
A E3uan 13 Tni. 3
0.5 1 1.5 2 2.*
Figure 1. Relationship between original AH and AHNEw as
a function of Re for trials 2 and 3.
Figure 1 is typical of the results obtained and the trends
observed from Eq. (11) for all the trial cases. At low Re
the data are close to the value 6 and at higher Re the data
approach asymptotically a value that is approximately
equal to 3. The difference in plots similar to the one of
Figure 1 is the approximate values of Re where the curve
flattens and becomes almost asymptotic. This is an
indication that another parameter is needed in the
functional representation of AH vs Re, which is similar to a
scaling factor. After several tests and calculations, it was
decided that the best parameter to be used as the scaling
factor is the Stokes number, St, defined in Eq. (9). The
acceleration number, Ac, which was used in the original
expression did not appear to be a good parameter for the
correlation, because its range in an oscillatory flow is from
0 to oo. The very large values of Ac result in significant
"spikes" in the computed values of the total hydrodynamic
force in the vicinity times given by the expression: ot= n7i,
with n being an integer. An inspection of the original
publication by Odar and Hamilton (1964) confirmed the
absence of any "spikes" in the graphs of the experimental
data. For this reason, St, which has a finite and relatively
narrow range of values, has been used instead of Ac in the
revised expression.
With the aid of the software TableCurve 2D we used the
computed values for AHNEw, derived from Eq. (11), to
derive a correlation AHNEw=f(Re, St). The best fit of the
computed data resulted in the following correlation:
AHNEW = 6.00 3.16 exp 0.14 ReSt 082)2] (12)
Accordingly, a semiempirical transient equation of motion
for particles was put together using the Odar and Hamilton
reconstructed data for the history term and the analytical
and empirical equations that have been proven correct for
the other terms. This new expression is as follows:
F = (i +0.15Re0687" ,p )2, p 1 '4 + (13)
Re 23 ) at
d(u, v,)
A ' )1' dt' dt'
where the new coefficients for the history term are
obtained from Eq. (12). Using this new correlation,
several more trials were run to compare the total
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
hydrodynamic force calculated using Eq. (2) to the total
hydrodynamic force calculated using Eq. (13). We
present these comparisons in the following paragraphs.
Results similar to the ones depicted in Figure 1 showed
that there is a significant difference between the original
and the new value of the history term coefficient.
However, there is almost no difference in the results for the
total instantaneous hydrodynamic force that are computed
using the original and the new expressions for the total
transient force. This is shown in Figure 2, where the
calculated results from the two expressions are shown for
the conditions identified as trials 2 and 3 in Table A. It is
observed in Figure 2 that the results for the total transient
hydrodynamic force are almost identical over more than
oneandahalf cycles.
200
.2O
1O nq h 2 l Snal
150 U l t "
200, E Tnml
Figure 2. Total instantaneous hydrodynamic force versus
time for trials 2 and 3 for the original and new
computations.
The relation seen in figure 2 shows a negligible difference
in the total hydrodynamic force that is exerted by the fluid
on the solid sphere. This is an indication that the
calculated terms for the new history term coefficient and
the reconstructed relationship that is depicted in Eq. (13) is
faithful to the original data by Odar and Hamilton. Thus,
the new expression, which is derived from these
calculations, is also faithful to the original data.
A good way to determine the agreement of two expressions
is to calculate the fractional absolute difference between
them. In this case the fractional absolute difference, FAD,
is expressed as a percentage and defined as:
FAD FINEw
FAD=100 '
IF.
Figure 3 depicts the FAD using data from trial 3. It is
apparent in the figure that the absolute fractional difference
in the calculated results from the original and the new
expressions is very low, typically less than 5%. This is
within the limits of the experimental error of the original
data, which was of the order of 10%. It is also observed
that the highest fractional difference in the computations is
in the vicinity ot= n7i (n is an integer), where the
acceleration term in the original expression, Ac, becomes
very large. This is another indication that Ac may not be
the best parameter to use in the correlation function of the
transient hydrodynamic force.
Paper No
10 ,
a ? * * * .* . .
0 2 4 6 8 10 12 14
Argument. rd
Figure 3. Absolute fractional difference of the total
instantaneous hydrodynamic force resulting from the
original Odar and Hamilton expression and the new
semiempirical expression under the conditions of trial 3.
Comparisons of the data from the other trials similar to the
ones shown in figure 3 have shown that the new
semiempirical expression for the transient hydrodynamic
force faithfully reproduces the results obtained from the
original Odar and Hamilton correlation. In addition, the
new correlation is in agreement with analytical and
experimental data for the added mass term and, as the
original correlation did, it is also in agreement with the
experimental data pertaining to the steady drag.
Therefore, the new correlation shown as Eq. (13) would
include a more accurate representation of the effects of the
history term than the original equation.
Conclusions
The Odar and Hamilton experimental data set is most likely
the best set of experimental data on the transient
hydrodynamic force on solid particles. However, the
expressions derived in the original study suffer from the
drawback of an erroneous assumption on the functional
form of the added mass term. A set of the data for the total
hydrodynamic force was reconstructed, following the
original instruction and calculations in the Odar and
Hamilton (1964) study. From the reconstructed data, a
new correlation for the history term was derived, which
more accurately represents the functional form of the history
term. The analysis showed that the new history term is
better correlated with the Reynolds and Stokes numbers,
rather than the acceleration number, Ac. The total force
acting on a sphere accelerating in a viscous fluid can be
accurately calculated by the new transient force
semiempirical correlation at Reynolds numbers in the range
0
the particle.
Acknowledgements
This work was partly supported by a grant from the DOE,
through the National Energy Technology Laboratory,
(DENT0008064) Mr. Steven Seachman project manager;
and by a grant from NSF (HRD0932339) Drs. Demetris
Kazakos and Richard Smith, project managers.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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