Microscopic Aspects of Turbulent Transport –Conduction and
Convection Unification
S. Usman§ , S. Abdallah*
And N. Katragadda*
§Department
*
of Mechanical Industrial and
Nuclear Engineering
University of Cincinnati, Ohio
Department of Aerospace Engineering
and Engineering Mechanics
University of Cincinnati, Ohio
Abstract: It is argued that transport of a scalar under the influence of turbulence is also
influenced by the molecular properties of the scalar. This argument is supported by the
observation of fluctuating velocity suppression during heavy gas cloud passage. Based on
this assumption it is possible (purely on mathematical grounds) to represent a problem of
simultaneous convection-diffusion by an expression identical to conduction equation using a
variable conduction coefficient. The mixed convection-diffusion transport equation is
linearized and rewritten in diffusion form. The new form of the transport equation combines
the convection and diffusion terms in a manner similar to the heat conduction equation with a
coefficient analogous to thermal conductivity. However, the coefficient developed in the
mathematical manipulation is a local variable which is a function of local diffusion and local
linearized velocities. It is shown that for diffusion coefficient0 the effect of diffusion
vanishes, producing a convection dominant transport mode. On the other extreme, for large
(infinite diffusion coefficient) the variable coefficient reduces to “thermal conductivity”,
producing pure diffusion equation.
Key-Words: - Conduction, Convection, Numerical stability, Diffusion, Turbulence
result molecular diffusion is a function of
both the fluid properties and the particle
characteristics (e.g. Hirschfelder et al,
1954). Significant advances have been
made to represent mutual molecular
diffusion coefficient of two gases which
consequently becomes a function of the
molecular/atomic properties of both the
gas molecules. (Chapman and Cowling,
1970).
Introduction
Analysis and modeling of transport of
passive scalar in a fluid medium has been
an area of research interest for the last 175
years or more. In 1827, Robert Brown – a
botanist observed perpetual random
motion of particles immersed in a fluid.
This observation has lead to the
development of the well known and
generally accepted molecular kinetic
theory of diffusion.
Displacement or
dispersion rate of a scalar under molecular
transport is accepted to be caused by
interaction of the diffusing particle with
the surrounding fluid molecules and as a
For fluid in motion, it is assumed that the
molecular movement is super-imposed on
the bulk transport of the fluid and if a
secondary fluid (for example a dye) is
introduced into a laminar flow, in addition
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to being transported along with the bulk
flow, random molecular movements also
diffuse the dye both along the flow and
also in the transverse directions.
Generally, it is believed that as long as the
flow is strictly laminar, the transverse
direction diffusion is solely due to
molecular diffusion (Treylab, 1980).
This is a very fundamental question with
far reaching consequences.
It is argued that in the absence of any other
mechanism, the exchange of momentum
and energy between the bulk fluid
molecules (even in turbulent motion) and
the diffusing particle is via molecular
collision and scattering. There are two
very significant aspects of this argument;
However, when the bulk flow becomes
turbulent a significant enhancement in
diffusion rate is observed (Hinze, 1959).
This turbulent diffusion is assumed to be
super-imposed on the molecular diffusion.
Turbulent diffusion becomes the dominant
contributor to the dispersion process and in
many cases molecular diffusion is
considered to be negligible (Taylor, 1921).
Random velocity fluctuations, which are
characteristic of turbulence, are believed to
be responsible for the enhanced turbulent
diffusion. Scales of these fluctuating
velocities are much larger than the
molecular motions. Generally, in the case
of turbulent diffusion modeling most of
the emphasis is placed on the flow
characteristics (that is the behavior of
fluctuating velocities) and the turbulent
diffusion rates are estimated without much
consideration given to the microscopic
characteristics of the molecules or particles
getting diffused in the flow.
1) If both the molecular and turbulent
diffusions are the result of the same
fundamental process, that is
molecular collisions and scattering,
are we justified in treating them as
independent and additive processes
as the commonly used approach
(see any standard text on
turbulence and viscous flow, e.g.,
White 1991)?
2) Is the use of an “eddy viscosity”
and “turbulent diffusivity” terms
which depend only on the
molecular viscosity of the bulk
fluid and the flow characteristics
justified, without due consideration
given to the microscopic nature of
particles or molecules getting
diffused?
Saffman (1960) introduced the concept of
interaction between molecular diffusion
and turbulence and concluded that the
interaction
can
potentially
reduce
dispersion. There is lot to be gained from
this classical work. Saffman’s analysis,
and almost all the other existing literature,
is based on the assumption that turbulent
and molecular diffusion are "independent
and additive" processes. His analysis did
not offer any insight into the dependence
of turbulent transport properties on the
molecular characteristics.
Let us pose a basic question, what is the
mechanism of energy and momentum
transfer from the bulk fluid to the diffusing
particle?
For the case of molecular
diffusion, it is well established that the
diffusion is a consequence of molecular
collision and scattering which is caused by
the random motion of the molecules
(Chapman and Cowling, 1970).
The
question at hand is, what is the
fundamental mechanism of energy and
momentum exchange between the particles
in turbulent motion and how the bulk fluid
molecules in turbulent motion transfer
their momentum and energy to the
molecule and/or particle getting diffused?
Heavy gas dispersion
Recently, a great deal of interest has been
placed on the atmospheric dispersion of
heavy gases (Hartwig, 1984).
Primary
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motivation for these studies is for
environmental impact analysis of pollutant
spills etc.
Several experimental and
theoretical studies have been conducted
pertaining to various aspects of heavy gas
dispersion in atmosphere (Davies, 1999).
In general, it has been observed that heavy
gases do not follow the expected rate of
diffusion in atmospheric air flow. Heavy
gas diffusion is much slower than expected
and a density-based stratification is
observed (Hunt et. al 1983 and Ooms &
Tennekes, 1984). These aspects of heavy
gas diffusion have made modeling of
heavy gas dispersion both a challenging
and interesting problem.
probability of scattering is uniformly
distributed) the average velocity becomes
proportional to the ratio m/M where m is
the mass of the incident particle and M is
the mass of the heavy target. Therefore,
when a puff of heavy gas is introduced in
the bulk flow, there is a suppression of
fluctuating velocity during the passage of
the heavy gas cloud. The extent of
suppression depends on the ratio of the
molecular masses and the initial velocity
of the heavy gas.
Likewise, in the other case when the spill
gas molecular mass is not significantly
different than the bulk fluid but the spill
gas is at a much lower temperature, most
of the K.E. transferred to the spill gas will
be used to achieve the thermodynamic
equilibrium and as a result the average
fluid velocity will reduce. As Mohan et. al
(1995) discussed, a transition to a true
passive transport will take place after a
duration of time which depends on the
atmospheric flow condition and the density
of the spill gas.
One aspect of heavy gas that has been
reported in the literature is the suppression
of fluctuating velocity during the passage
of the heavy gas cloud (Koopman et. al
1989). Heavy gas clouds can be classified
into two groups; ones that are made up of a
gas with heavy molecules and others
where the gas molecular weight is not
significantly larger than the bulk fluid but
is at a temperature/pressure that the
density of the spill gas is much larger than
the bulk of the fluid in motion. Or in other
words, the spill is not in thermodynamic
equilibrium with the bulk fluid. For both
of these cases the fluctuating velocity
damping can be explained using the
argument that the mechanism of energy
and
momentum
transfer
between
molecules in a turbulent flow field and the
spill gas is via molecular collision and
scattering.
The phenomenon of fluctuating velocity
suppression is observed and reported by
Koopman et. al (1989).
In the present analysis unification of
conduction and convection phenomena is
brought about by writing the well known
vorticity transport equation in a simple
diffusion equation form with a spatially
and
temporally
varying
diffusion
coefficient.
Conduction convection unification
Consider the case of atmospheric
dispersion of a gas with molecular mass
significantly larger than the average mass
of air (say N2) molecule.
For the
simplifying
assumption
of
elastic
scattering where both energy and
momentum are conserved, as the mass of
the target particle increases the average
particle velocity after elastic scattering
interaction reduces. In fact for the case of
isentropic scattering (that is, the angular
The steady state transport equation for two
dimensions (X,Y) can be written as;
 xx   yy 
1
(u x  v y )  0 (1)
D
where  is the transport scalar, ( u , v ) are
the velocities in (X,Y) directions
respectively, D is the diffusion coefficient.
This non-linear partial differential
3
equation is linearized and rewritten in a
diffusion form, the linearized form is;
Multiplying and dividing the first term in
v . y
equation (3) by e
1
1
 xx  u  x   yy  v  y  0
D
D
D
and the second term
u . x
(2)
by e
D
,
u and
v are
where
the
velocity
components at the location (x,y) in the
sub-domain.
Both u and v can be
considered local constants. Equation (2) is
written in the following integrating factor
form in (X and Y) directions;
u .x
e
D
v .y
 u . x 
 v . y 
D
D
.e
 x   e .e D  y   0

 x

 y
(3)
u .x
e
D

v .y
D
 
.e


u .x
D

v .y
D
u .x

x   e D

x

D
Or
K. x x K. y y 
0
(5)
Where
K e
(6)

u
v
x y
D D
To obtain a numerical scheme
Equation (5) is integrated over the
control volume defined by 5 point
domain shown in Fig. (1).
Equation (5) is similar to the heat
conduction equation, where the
dependent variable,  and the factor K ,
y / 2
 K
x x  x / 2
 K x
  u .x  v . y 
D
.e D
 y   0 (4)



y
represent the temperature and the
thermal conductivity respectively. It
should be noted that here, K in
equation (6) is variable with respect to
(X,Y) the space variables. It may also
be noted that K can be estimated for
every field point in a flow field.
v .y
dy   K
x  x / 2
x   x / 2
y   y / 2
y y  y / 2
 K y
y   y / 2
dx  0 (7)
x   x / 2
equal to the intermediate values
between the grid points, Eq. (7) results
into the 5-point scheme
If  X and  y are taken to be constants
on the control volume surfaces and
b i 1, j  b i 1, j  b i , j 1  b i , j 1  b i , j
(8)
Where

bi 1, j  Exp p  q   Exp p  q / 2x q
(8a)
2
bi , j  bi 1, j  bi 1, j  bi , j 1  bi , j 1

(8b)
p
(8c)
4
ux
2D
q
vy
2D
solutions are computed using VAX/VMS.
The following two cases are considered
(8d)
First case: Re u  Re v  0.01
The rest of the coefficients in Eq. (8) can
be obtained by permutation. x and y are
the space increments in X and Y directions
respectively. Equation (8) was previously
derived, consequently no further testing
for the 5- point scheme is considered in the
present work.
Second case: Re u  100 and Re v  1 .
Boundary conditions:


 



 

 ( x,0)  1  Exp Re u ( x  1) / 1  Exp(u Re)
 (0, y )  1  Exp Re v( y  1) / 1  Exp(v Re)
Numerical Results
Equation (8), the finite difference form of
Eq. (1) is solved at each grid point of the
square domain using the Successive OverRelaxation method (SOR). The relaxation
factor is taken as unity. The numerical
 ( x, y ) 
 ( x,1)   (1, y )  0
Exact Solution is given by:
1  ExpRe u( x  1)1  ExpRe v( y  1)
1  Exp(u Re) 1  Exp(v Re) 
gas/particles is molecular collision and
scattering. This assumption raises several
basic questions very pertinent to turbulent
transport phenomenon.
The numerical results for both the cases
are obtained using equal space increments
in X and Y directions (x  y  0.10) .
The results of the first case are computed
in about 3 CPU seconds, Fig. (2). This
case is considered here in order to check
the validity of the present method at low
cell Reynolds numbers. The results for the
second case are given in Fig. (3) in
comparison with the exact solution. These
results are obtained in lesser iterations than
the first case.
It is shown here that a convectiondiffusion equation can be represented as a
pure diffusion equation with a variable
conductivity. That variable conductivity
coefficient is a function of the convection
velocity and the molecular diffusivity. In
other words, the dispersion of a scalar
quantity is a diffusion process with
diffusion coefficient an exponential
function of the molecular diffusion
coefficient and the convection velocities. It
is important to point out that the exponent
in the variable conductivity in equation (6)
is a function of the local properties versus
the global property, the diffusion
coefficient D . The analysis presented by
equations (5) and (6) is applicable in
general for flow momentum equations, the
Discussion and conclusions
Based on the observation made by
Koopman et. al (1989) and the argument
presented here it seems logical to assume
that the fundamental mechanism of energy
and momentum transfer between the bulk
fluid molecules and the diffusing
5
energy equations and the vorticity
equations. The implication for momentum
and vorticity equations is that the
exponential conductivity, K is the cell
Reynolds number (Peclet number). This
scheme leads to a positive conductivity
coefficient and a diagonal dominance of
the discrete formulation for numerical
solution of equation (5) for all Peclet
numbers versus direct solution for
equation (1). Diagonal dominance has the
obvious advantage of numerical stability
and elimination of oscillations resulting
from non-physical phenomenon. In other
words, equation (5) filters non-physical
References
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Mathematical Theory of Nonuniform
Gases, 3rd Ed., Cambridge Press, London.
Hartwig S., (1984) Heavy Gas and Risk
Assessment III : proceedings of the Third
Symposium on Heavy Gas and Risk
Assessment, Bonn, Wissenschaftszentrum,
November 12-13.
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Stratified Flows (5th 1996 : University of
Dundee) Oxford , Clarendon.
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Hirschfelder, J.O., Curtiss, C.F., & Bird R.B.,
(1954) Molecular Theory of Gases and
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oscillations in equation (1) of the cases of
large Peclet numbers.
The proposed approach of unifying
diffusion and convection is coherent with
Yakhot’s (2003) suggestion that the
dissipative scale is much smaller than
Kolmogorov’s (1941) scale.
In other
words, there can potentially be continuity
between molecular scales and the turbulent
scales. It is only that these very high
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extremely small length and time scales are
impractical to produce using conventional
mechanical means.
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