JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER
Vol. 6, No. 2, April-June 1992
Two-Dimensional Radiative Transfer in a Cylindrical Layered
Medium with Reflecting Interfaces
N. M. Reguigui* and R. L. Doughertyt
Oklahoma State University, Stillwater, Oklahoma 74078
A system of exact linear integral equations for the source function, intensity, and flux is presented for a twodimensional cylindrical medium consisting of up to four layers with reflecting interfaces between the layers.
Properties that may change from layer to layer are the single scattering albedo, optical thickness, and refractive
index. The incident radiation is coUimated and has a Bessel function distribution. The Bessel function boundary
condition reduces the two-dimensional problem to a one-dimensional problem. Superposition is then used to
derive the solution for any other boundary condition that is Hankel transformable. A special case of a Gaussian
distribution that models a laser beam is presented. Some one-dimensional numerical results are presented for
the source function and intensity within one- and two-layer media. Two-dimensional results are presented for
back-scattered intensity due to the laser-beam boundary condition. Only the conservative case, optical thicknesses
of 2.0 and 5.0, and refractive indices of 1.00 and 1.33 are considered.
Zt
z
j3
f$e
AT,
8
8fj
O
function defined in the Appendix
physical depth coordinate
Hankel transform parameter
extinction coefficient
optical thickness of layer i, rzi - rzi_l
Dirac delta function
Kronecker delta function
angle between the incoming and scattered
radiation
6
= polar angle
Aiy = kernel function
Mext = cosine of the polar angle outside the medium
fjii
= cosine of the polar angle, 6-, within layer i
Miy.cr = cosine of critical polar angle between layers / and j
IJLO = cosine of incident polar angle of intensity
&J
= function defined in the Appendix
Pij
= Fresnel's coefficient of reflectivity for energy
crossing the interface between layer i and layer j
r0
= overall optical thickness of all layers of a given
medium, rzN - TZO
Tr
= optical radial coordinate, fier
Tro = laser beam optical radius, fijr0
rs
= optical coordinate along the general threedimensional s direction
TZ
— optical depth coordinate, pez
rzi = optical depth at the bottom of layer i
Tzi_l = optical depth at the top of layer /
<&ij - function defined in the Appendix
</>
= azimuthal angle
\lft
= function defined in the Appendix
ft
= solid angle
(Oi
= single scattering albedo in layer /
Nomenclature
A,
B{J
DN
E
Giy
g
/,
/+
It~
I0
IT
/,
J0
Kfj
W
HIJ
P
Qij
qzi
Ri
r
r0
5,
Scio
T
Ti
tfj
x
= multiple reflection coefficient, defined in the
Appendix
= function defined in the Appendix
— function defined in the Appendix
= component of kernel function for source function
integral equation
= function defined in the Appendix
= Hankel transformed function
= intensity of radiation in layer i
= intensity in layer i in + TZ direction
= intensity in layer i in — rz direction
= magnitude of incident intensity outside the
medium
= intensity transmitted across an interface
= magnitude of incident intensity
= zeroth order Bessel function of first kind
= function defined in the Appendix
= total number of layers
= ratio of layer i refractive index to that of layer ;
= scattering phase function
= function defined in the Appendix
= z direction flux in layer i
= function defined in the Appendix
= radial distance from center of medium
= laser beam radius
= source function in layer i
= source function lead term, defined in the
Appendix
= transmission function, defined by Eq. (32)
= function defined in the Appendix
= transmission through the interface between layer i
and /
= dummy integration variable
=
=
=
=
=
=
=
=
Subscripts
cr
ext
i
j
r
5
Presented as Paper 90-1779 at the AIAA/ASME 5th Joint Thermophysics and Heat Transfer Conference* Seattle, WA, June 18-20,
1990; received Nov. 20, 1990; revision received March 25, 1991; accepted for publication April 2,1991. Copyright © 1990 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
* Graduate Student, School of Mechanical and Aerospace Engineering, EN 218.
tAssociate Professor, School of Mechanical and Aerospace Engineering, EN 218. Member AIAA.
related to critical angle
external to all layers of the medium
layer i
layer;
related to radial direction
related to a general three-dimensional direction
related to depth direction
Hankel transformed variable
Superscripts
1,2 = either function 1 or 2 (not both at once)
+ , - = either positive ( + ) or negative ( - ) z direction
(not both at once)
232
233
REGUIGUI AND DOUGHERTY: TWO-DIMENSIONAL RADIATIVE TRANSFER
Introduction
I
N radiative transfer problems, many realistic situations,
such as solar pond modeling, laser-material interaction,
and underwater radiation back scattering, require a multidimensional analysis that accounts for the effects of the refractive index. However, most of the results found in the literature
employ many simplifying assumptions, such as one-dimensional geometries or multidimensional geometries that neglect
the effects of the index of refraction and consider only one
or two aspects of the radiative transfer problem of interest:
1
reflecting boundaries,1"4 diffusely
incident radiation
and col4
5
limated incident radiation, isotropic scattering and anisotropic scattering,6 variation in absorption and scattering with
depth,7-8 and two or three dimensionality.5-6-8 A more realistic
analysis has to include the effects of multidimensionality as
well as the effects of possible variations in the pertinent physical properties, such as index of refraction, and absorption
and scattering coefficients of the medium.
The index of refraction change across an interface, coupled
with the directional redistribution of radiation by anisotropic
scattering, has been shown to alter considerably the radiative
field within the medium.1 Cengel and Ozisik2 presented onedimensional data for a one-layer slab which indicate that the
constant reflectivity approximation often leads to severalfold
under- or overestimated results as compared to the exact handling of the reflection coefficient; thus disproving the assumption of using average values.
Most of the work that includes the effect of index of refraction has been limited to one dimension. Buckius and Tseng1
considered a plane-parallel isothermally emitting medium that
scatters anisotropically. Armaly and Lam3 and Dougherty4
have investigated the influence of refractive index on the reflectance from a semi-infinite one-dimensional absorbingscattering medium.
Although a lot of work has been done for the two-dimensional geometry, index of refraction effects have been neglected for much of that work. A two-dimensional problem
that has received attention in the past 10 years is the back
scattering of a laser beam by a multiple scattering medium
when the incident beam is normal to the surface of the medium.5'6 Most of these studies were based on the assumption
that the index of refraction is unity. Crosbie and Dougherty
presented graphical and tabular results for isotropic5 and
anisotropic6 scattering in a finite cylindrical medium exposed
to a laser beam. This present work builds on these studies to
include the effects of the index of refraction and the effects
of layering the medium.
Some investigators9-10 have considered superposing the unit
refractive index results or introducing a factor in the results
based on the unit index of refraction to account for index of
refraction effects. Crosbie and Dougherty9 have proposed superposing the unit refractive index results to obtain directional
and hemispherical reflectances from a two-dimensional cylindrical medium. No numerical results were presented, but some
asymptotic solutions were examined. In a comparison between theoretical and experimental results for back scattering
from a two-dimensional optically thick medium exposed to a
laser beam, Nelson et al.10 have found that agreement between theory and experiment is improved when the theoretical back-scattered intensity for unit refractive index is reduced by (1 - pN)2ln2. However, this approximation was
found to be valid only in the thin limit.
Studies of radiative transfer in composite (layered) media
including the effects of scattering are very limited and mostly
deal with one-dimensional situations. Shouman and Ozisik11
considered the problem of radiative transfer in an absorbing,
emitting, and isotropically scattering two-layer slab with diffusely and specularly reflecting boundaries. The slab was irradiated externally by diffuse radiation at the boundary surface. The FN method was used to compute results for the
transmissivity and reflectivity of the slab. The results showed
that by increasing the single scattering albedo of the second
slab, the reflectivity and the transmissivity of the composite
slab increase. The transmissivity was found also to be influenced by the relative optical thicknesses of the slabs.
A few other studies also exist for media with layered properties; but usually the index of refraction is assumed to be
unity, which considerably simplifies the analysis. Stamnes and
Conklin7 have developed a matrix method to solve the discrete
ordinate approximation to the radiative equation in a homogeneous, plane parallel atmosphere consisting of N adjacent layers. However, because of the one-dimensionality of
the analysis and the unit index of refraction, this study is
applicable to the field of atmospheric physics and meteorologyA general multilayer study was done by Sutton and Kamath.8 They considered a three-dimensional rectangular medium with layered properties and unit index of refraction. The
boundaries between the layers were considered transparent,
hence neglecting the effects of index of refraction changes.
The volume-averaged discrete ordinates method was employed to solve for the transmitted and reflected radiative
flux.
In the present investigation, the exact solution for the radiative transport equation within an absorbing and isotropically scattering two-dimensional cylindrical medium is presented. The medium is finite in height and infinite in the radial
direction, has internally reflecting top and bottom surfaces,
and consists of layers whose single scattering albedo, optical
thickness, and index of refraction may differ (see Fig. 1). The
interfaces are assumed to be smooth so that Fresnel's equation
and Snell's law can be used. The incident radiation is collimated and normal to the top surface of the medium and it is
assumed to be azimuthally symmetric. It has a radial variation
in the form of a Bessel function. The Bessel function boundary
condition is used to separate variables, and therefore reduces
the two-dimensional equations to one-dimensional form. Superposition is then used to solve the problem in which the
incident radiation is in the form of a Gaussian function (i.e.,
a laser beam). No radiation is incident on the bottom surface
of the medium.
Development
The transfer equation in a given coordinate system for a
certain layer i in a medium consisting of many layers and in
which scattering and absorption take place is5
=
/,(T.)P(cps 6) dft
(1)
Io
Tr-XX>
CDI
Fig. 1 Geometry of a layered medium, infinite in the rr direction and
finite in the TZ direction.
REGUIGUI AND DOUGHERTY: TWO-DIMENSIONAL RADIATIVE TRANSFER
234
In the source function 'S-h a>, is assumed to be constant for
layer -i, but could differ among the layers. For the remainder
of the analysis, P will be assumed equal to one for isotropic
scattering.
Decomposing Eq. (1) into its positive (7+) and negative
(/") directions, and then solving both equations using an integrating factor approach, yields5
x exp[(T2l_! - T2)//xf] +
exp[(r; Jrzi-l
the intensities at the interior interfaces between the different
layers are unknown, and they need to be eliminated from the
set of Eq. (6) in order to solve for the source functions for
all layers.
Intensities and Source Functions
We will consider collimated incident radiation at the topmost boundary. Thus, the radiation crossing the interface of
an arbitrary layer i has, in general, a collimated part, due to
the collimated incident radiation, and a diffuse part due to
scattering from the adjacent layers. The collimated radiation
can be represented using the delta function, i.e.
(2)
(7)
and
(3)
/; is the magnitude of the intensity and (/t/,0) determines the
direction of the collimated radiation. From the law of conservation of energy, the relationship between the incident and
the refracted intensities is3
(8)
where r'r is given by
(T;)2 =
(4)
with
«2 = Tl = (r'z - r2)2(l - M ; 2
(5)
and r'nf . = r, for t2 = r2l (see Fig. 2). All of the quantities
with subscript i are related to layer i, and rzi and Tzi_l refer
to the bottom arid top of the layer i, respectively (see Fig. 1).
5 is a function of rr and rz in Eqs. (2) and (3) only because
of the isotropy of the problem and the axisymmetric loading
assumption. ^(r^r^.^p.^) and 7f (T^T,,,/^) are the
boundary conditions at the top and bottom of layer /, repectively, which is finite in the depth (rz) direction and infinite
in the radial (TV) direction. Substituting Eqs. (2) and (3) into
the right-hand side of Eq. (1) and rearranging the result gives
the integral equation for the source function in layer /
My = [1 - (1 -
(9)
The imaginary part of the refractive index is assumed to be
small relative to the real part, which is true for most dielectrics. When n^ > 1, Eq. (9) cannot be satisfied for ^ < ju,/7cr
where
/4cr
(10)
Therefore, when niy > 1, all intensity that is coming at angles
Of greater than 0£> cr (or ^ < ^ cr) will be totally reflected (#
= 1.0).
When the intensity crossing the interface between layers i
and j is diffuse,
the amount of the refracted intensity in medium j is3
(11)
x exp[(Tz|._, - Tz)/p.'t] + /r(T;.,TZI.,/i>')
x exp[-(r2/ - T2)/M/]} dM; d<£' + ^ P* P P*
477 JO
where 7y is the magnitude of the intensity refracted into layer
;'. The reflectivity function p,7 is given by Fresnel's equation1
and /Xy is given by (SnelPs Law1)
JO JTZ/-I
(6)
Equation (6) actually represents a set of equations to be solved
for all layers in the medium. Besides the top and bottom
boundary conditions of the medium which should be known,
In Eq. (2) /^(T^_i,T z/i _ !,/*,-,<£) is the sum of the incident
radiation that crosses the interface and the radiation originating in the layer that is reflected at the top interface in the
positive direction. Therefore
(12)
Similarly, the boundary condition /"(T^T.,,,/^,^) in Eq. (3)
is the sum of the interface transmittance of radiation from
layer / + 1 plus the interface back reflection of scattered
radiation at the bottom of layer /. Therefore
(13)
Fig. 2 Cylindrical coordinate system.
In Eq. (12), /r( T ri-u T z/-i?M'/^) is. replaced by Eq. (3), where
TZ is set equal to r2l_!. The transmitted intensity
Jr( T r/-i> T z/-i>A t /><£) could be collimated and/or diffuse. Similarly, in Eq. (13), /^(r^r^,^,^) is written from Eq. (2)
where TZ is set equal to r2l and /f (r^,T2/,^/,<^) is determined
from Eq. (8) and/or Eq. (11). Both resulting equations can
be solved simultaneously for /^(r^.^T^-.!,^,^) and
Irfr'rhTziMd) [using Eqs. (2) and (3) a second time in the
process]. This will result in a system of linear simultaneous
REGUIGUI AND DOUGHERTY:
TWO-DIMENSIONAL RADIATIVE TRANSFER
equations in terms of the interface intensities. When the solution to this sytem is put back into Eqs. (2) and (3), we get
235
Equation (16) represents a system of simultaneous integral
equations in terms of the source functions of each layer for
a cylindrical medium consisting of N layers. 5f0(/^l,r2) is an
initial term due only to the collimated boundary condition,
and it is being evaluated at //,,- = 1 because of the normal
incidence of the radiation [see Eq. (Al)]. Again, //,, should
be greater than the critical angle /x,/y cr when considering radiation from layers other than layer i. The auxiliary function
T/y is another collection of terms due to the transmission,
reflection, and attenuation of the radiation as it crosses the
different layers, and it depends on the relation between / and
; as given in the Appendix [Eqs. (A16)-(A18)]. Note that
Eqs. (14), (15), and (16) reduce to those of Buckius and
Tseng1 when there is only one layer (except that their boundary condition is diffuse while this boundary condition is collimated).
Radiative Flux
Once the intensity of radiation is determined, the radiative
flux in the z direction may be computed from substituting
Eqs. (14) and (15) into the following equation:
P" Stt.TiWO^T;) dr'
JTzi-1
(14)
and
~i
(i?)
Having the basic two-dimensional equations derived for the
collimated boundary condition, it is easy to write those equations for other forms of boundary conditions. This is explained
in more detail in the next section.
Special Boundary Conditions
/=!
Bessel Function Boundary Condition
exp[(r, S,(T;,T;)*KM»,T;)
(15)
where because of the axisymmetric loading, the intensity does
not depend on </> directly, except through rr. The equations
for AX/0, DM, Z+GO, Zr(/0, *,W Rft*,), 77fo),
T?(/ij), BJO^Tv), Bl{ni9Tr), *ft(Hi,Tz)9 and ttf(Hi,Tz) are given
in the Appendix. The first lead term in each of Eqs. (14) and
(15) has an infinite value when IJLO = 1 due to the Dirac delta.
The remaining terms in the equations include the radiation
contributions from all the layers other than layer /, as well as
the contribution from within the layer. Notice that when the
number of layers is greater than one, r'r will be given by a
more general formula than Eq. (4), i.e., Eq. (A28).
In the process of obtaining Eqs. (14) and (15), the following
equation for the source function was obtained:
Even for the isotropic case, the two-dimensional equations
are still difficult to solve. However, if the incident radiation
is assumed to be in the form of a Bessel function, then the
source function, the intensity, and the flux can be shown5 to
be separable functions of rr and r2, and the two-dimensional
governing equations can be reduced to simpler one-dimensional forms. The Bessel solution can then be superposed to
obtain the solution for any Hankel transformable boundary
condition.5
A Bessel function boundary condition has the following
form:
(18)
where /, is a constant, the magnitude of the intensity.
Source Function
The Bessel-varying boundary condition suggests that, for
this case, the source function can be separated,5 i.e.
(19)
and S0,XTz) represents the j3-varying one-dimensional source
function for layer /. Substituting Eqs. (18) and (19) into Eq.
(16) and then, using a technique similar to Crosbie and
Dougherty,6 the /3-varying source function can be found12'13
(20)
REGUIGUI AND DOUGHERTY: TWO-DIMENSIONAL RADIATIVE TRANSFER
236
where
a),
- r'z) + f
/.(flr
•rpftcr
Jrzi-l
(21)
(26)
-(T; - Tz)/M,]5p,(r;) dr'z
where raij is defined by Eq. (A29), and where it was necessary
to split the integral over /*/, because when/ is different from
i, /Lty will be restricted to the interval between /*l>fCr [given by
Eq. (10)] and 1.0. E in Eq. (21) is a Bessel-varying exponential
integral, and it is defined as
Flux
Similarly, the flux can be written as a separable function
of Tr and TZ, i.e.
(22)
(27)
which is related to the generalized exponential integral of Ref .
5. This integral reduces to the exponential integral when ft is
zero (the one-dimensional case). I\ can be numerically computed for any given values of /i,', TZ, and r'z from Eqs. (A16)(A18). Then Eq. (21) can be integrated numerically to compute A/y, and this can be used in Eq. (20) to solve for the /3varying source function.
Substituting this equation and Eqs. (23), (25), and (26) into
Eq. (17), and following a similar development as for the source
function gives
wto =
[exp[(T,,_, - T,)]Z,+(
Intensity Equation
The Bessel-varying boundary condition suggests that the
intensity can be separated,5 i.e.
(23)
,,.,, - T, - Any/*,'] -
where /^,/(TZ,^,<^) represents the /3-varying one-dimensionall
intensity
for layer i. Similarly, the effective intensity terms B
and B2 [of Eqs. (14) and (15), defined in Eq. (A19)] can be
separated as
]
i-l
+ 2
/=!
(24)
}
where B^j(fit) is now defined in terms of the integral of Spj(rz),
as in Eq.' (A20). Substituting Eqs. (18), (23), and (24) into
Eqs. (14) and (15) gives the j8-varying intensity equation, viz.
(x)
f 1 [Tzi
d/Lt/ + Y Jo JTZ
i
sgn(rz - T'Z)
.n
x exp(- An/ft) 2
ttflj/M
.
AM)
drz
+
where Bqft is a modification of the function
(A20), replacing S^r'z) by S,,(r'z)J0(ftra^
- T^J^KM.) f" S,XT;)W(MI,T
ft9
given in Eq.
Other Boundary Conditions
JTZ/-I
(25)
and
(28)
After the basic solution for the Bessel function boundary
condition has been found, this can be used in superposition
to find the solution for any other boundary 5condition that is
expressible in terms of a Hankel transform. Representation
of the rr-varying boundary condition, /0(rr), by a Hankel
transform yields5
exp[ - (rr/ I0(rr)
(29)
- 1) +
The solution of any other problem with a boundary condition
that is Hankel transformable can be found from an inversion
process similar to Eq. (29)—substituting the source function
or flux for intensity as required.
237
REGUIGUI AND DOUGHERTY: TWO-DIMENSIONAL RADIATIVE TRANSFER
Gaussian Distribution
In this work, a Gaussian functional form is used to model
a laser beam,5 such that I0(rr) and is
/.(TV) =
(30)
where rro is the effective optical radius of the beam (at the
e~* point). The source function inversion equation becomes12
(31)
where the integration over /3 has been replaced by integration
over x ( = /3Tr), and rjrr has been replaced by rjr (assuming
uniform properties within each layer). The corresponding intensity and flux equations are similar to Eq. (31).12-13
sults. Tables 1 and 2 present a comparison of results generated
by this multilayer program with those of Dougherty.4 In those
tables, "Direct" and "Ambar" refer to solution methods employed by Dougherty, solving the integral equation for the
source function by discretizing rz (Direct), and solving only
for the source function at the boundary by a variation of
Ambartzumian's method (Ambar). Table 3 gives a comparison between the current results and those of Dougherty14 for
a one-layer medium with both boundaries having reflective
interfaces. Current 1L and Current 2L refer to employing the
modeling of this paper, and using either one uniform layer
(1L) of a given number of rz quadrature or two uniform layers
(2L) having the same properties, but twice the number of rz
quadrature. In addition, Table 3 uses T(p), which is given
by12
(32)
Results
Numerical Procedure
The method of solution consists basically of using Gaussian
quadrature to do the different integrals involved, and applying
successive approximation to determine the source function of
Eq. (20). First, the different A/y(j8,rz,rO values given by Eq.
(21) are computed at the quadrature points TZ and T'Z for every
P. These values necessitate a large amount of storage, about
one-third megabyte per )3 for a one-layer problem. In addition
to storage, the computations require approximately 45 CPU
minutes on an IBM 3090 per /3 calculation for a one-layer
problem. Increasing the number of layers obviously increases
this computational time. But once the Al7 values are computed, the iterations to compute 5^ become quite fast. Also,
since the A/y values do not depend on albedo (w), the source
function can be determined for a variety of a) values without
having to evaluate the Aiy values for every (o. To start the
iteration on Eq. (20), S is first set equal to 5f0(l,rz) of Eq.
(Al).
Test runs have shown that the function riy(//>,Tz,T^)vhas several irregularities in the curve as a function of the cosine of
the angle p. This is due to the sharp change in the reflectivity
coefficient function1 whenever the argument (a function of
ju,) becomes larger than the appropriate critical angle. To
alleviate this problem, the integral was divided into several
subintervals determined by all of the critical angles that can
exist when going from layer / to layer;; and a dense quadrature
was placed around each of these breakpoints. This procedure
increased the execution time considerably, but on the other
hand, helped to assure the accuracy of the results. Convergence in computing the source function was achievable for
any accuracy desired and for any set of variables (w,n,AT, . . .).
However, for albedo equal to unity, the convergence was
relatively slower than for other values.
To test the accuracy of the computer program, some published results were first reproduced. Then, some particular
two-dimensional multilayer problems were considered. Most
of the cases from the literature that have been considered are
in good agreement with the generated results. Unfortunately,
all of those cases deal with one-dimensional geometries, as
there is no other work in the open literature that considers
index of refraction effects for two-dimensional geometries.
One-Dimensional Results
Two main problems were considered for comparison. The
first one considers a single-layer one-dimensional medium
with a top reflecting surface.4 The second case has, in addition
to the first case, a bottom reflecting surface.14 Both cases have
collimated intensity incident on the top surface and no incident intensity on the bottom surface of the medium. By setting
)8 equal to zero and the number of layer (N) to one, the current
results were comparable to one-dimensional single-layer re-
In all of these tables, the maximum error between any two
entries is less than 0.07%. If the layer is divided into two
sublayers that have the same properties, then the two-layer
model (Current 2L) predicts the same values as compared to
those from the one-layer model (Current 1L) as shown in the
tables. The slight differences in results between the 1L cases
and the 2L cases are due to the increased number of quadrature for rz.
Some additional one-dimensional results will be presented
first in order to show the effects of changing the index of
refraction between two layers. The one-dimensional results
are a special case (/3 = 0) for the /3-varying solution. Figure
3 presents a plot of the source function as a function of the
medium depth (rz). The medium consists of two layers. The
top layer models a thin film of paraffin oil15 (n^ = 1.48) with
an optical thickness of 0.1. The second layer has an optical
thickness of 5.0 and an index of refraction of 1.33 (water).
The albedo is assumed to be unity for the water and 0.9 for
the oil. These albedos can be experimentally simulated by
adding small latex spheres to the fluids.15 The graph shows a
Table 1 Comparison of current results with those of Dougherty4
4077^/0 = o,i (0,/i)/
s~ _fnv
Direct4
Ambar4
Current 1L
Current 2L
'm(l)
1.486318
1.486321
1.485868
1.486116
4)i(l)
mf-i n
fji = 0.5
/Lt = 1.0
~ <?z0 = 0,l(0)]
2.72818
2.7282
2.72752
2.72772
4.25353
4.25366
4.25246
4.25710
0.60330
0.60331
0.60315
0.60323
«10 = 1.33, «12 = 1.00, TO = 6.0, to, = 0.5, 0 = 0.0.
Table 2 Comparison of current results with those of Dougherty4
Direct4
Ambar4
Current 1L
Current 2L
'oi(l)
/LI = 0.5
/Lt = 1.0
2.001094
2.001031
1.999625
2.00037
5.44812
5.44790
5.44448
5.44628
8.74655
8.74616
8.74038
8.74066
- ^=0,i(0)]
0.92609
0.92604
0.92544
0.92577
n10 = 1.50, «12 = 1.00, T0 = 8.0, o>! = 0.7, /3 = 0.0.
Table 3 Comparison of current results with those of Dougherty14
'oi(l)
Direct14
Current 1L
3.857896
3.80523
,
'm(l)
2.557394
2.556909
T(fJL
2.101162
2.100681
/i,o = 1.50, nu = 1.20, TO = 1.0, o>, = 1.0, 0 = 0.0.
1.965850
1.965385
REGUIGUI AND DOUGHERTY: TWO-DIMENSIONAL RADIATIVE TRANSFER
238
0.40
6.0
1 Oil/Water Layers
2 Water Layer, To = 5.0
3 Water Layer, ro = 2.0
5.0
(0.0, 1.0), TO= 2.0
(To, 1.0), TO • 2.0
(0.0,1.0), r0 : ' 5.0
(To, 1.0), TO = 5.0
0-0
4.0
CO
.
3.0
2.0
0.0
1.0
2.0
3.0
4.0
5.0
0.00
10-3 1Q-2 1Q-1
1.0
101
102
1Q3
ft
OPTICAL DEPTH (Tz)
Fig. 3 One-dimensional source function inside the medium (for oil:
at = 0.9 and n = 1.48; for water: o> = 1.0 and n = 1.33).
Fig. 6 Effect of optical thickness on the 0-varying intensity for o> =
1.0 and n = 1.33.
dramatic drop in the source function just at the interface
between the two layers. This drop is due to the change in
0.225
index of refraction from 1.48 to 1.33, which results in more
radiation being reflected back into the thin layer. The thin
layer exerts less influence on the source function in the medium as we move away from the interface. This is shown by
comparing curve 2—the source function for the same layer
of water (r0 = 5.0) but without the second thin layer—to the
i? 0.150
curve resulting from the addition of the thin layer. The third
curve (TO = 2.0, water only) on Fig. 3 is included to show
the effects of varying the optical thickness of the layer on the
source function. Small optical thickness means less scattering,
and hence a smaller source function.
1 I , water-oil
—
r. 0.075 Another special case of interest is the optical thickness and
q
2 I*, water-oil
the index of refraction effects on the angular distribution of
o
3 IT TO = 2.0, water _
the back-scattered intensity (7~) and the transmitted intensity
4 I* TO = 2.0, water "
i'5
(/+). The one-dimensional solution, which is still the special
5 n TO = 5.0, water "
case of j8 = 0, gives satisfactory insight into these effects.
6 I* TO = 5..0, water 0.000
Figure 4 gives the back-scattered
intensity (7~) and the trans0.00
0.20 0.40
0.60 0.80
1.00
mitted intensity (7+) outside the medium for the cases of oneand two-layer media. For the one-layer medium (n = 1.33,
CD = 1.0), comparing r0 = 2 and r0 = 5, the back-scattered
Fig. 4 The intensity outside a one-dimensional medium: one- and
intensity is increased for the larger TO. This trend is reversed
two-layer effects (for water only: w = 1.0; for water-oil: o> = 1.0
for the transmitted intensity. When a thin layer (r0 = 0.1 and
and TO = 5.0 for water, while o> = 0.9 and r0 = 0.1 for oil).
(o = 0.9) of paraffin oil is introduced on top of the water
layer (for the case where TO = 5), both transmitted and backscattered intensities are reduced drastically.
n = 1.33, TO = 2.0
2 80,1(0.0), n - 1.33, T0 = 5.0
3 S^j (T0), n = 1.33, TO = 2.0
4
S£,J(TO), n = 1.33, TO - 5.0
5 S^j(O.O), n = 1.00, TO = 5.0
6 S^j(T0), n - 1.00, r0 - 5.0.
O.OI___i_
10-3 1Q-2 io~1
_____________
1
2
1.0
10
10
103 104
Fig. 5 Effect of refractive index and optical thickness on the /3-varying source function for o> = 1.0. (Curves 5 and 6 are from Ref. 16).
Two-Dimensional Results
Before presenting some two-dimensional results, we will
examine the /3-varying solution. Figure 5 presents the influence of optical thickness TO, spatial frequency /3, and index
of refraction on the source function. The effects of these
parameters on back-scattered intensity and transmitted intensity are illustrated in Figs. 6 and 7. The source function at
the top and bottom of the medium exhibits similar trends for
both optical thicknesses. For small /3 values, the source function at the top is larger for the smaller r0. This trend reverses
for the bottom boundary source function. These functions
are
compared to the results by Crosbie and Koewing16 for unit
refractive index. It is found from the comparison that the
index of refraction increases the /3-varying results for the source
function, but reduces the intensity results (Figs. 5 and 7).
Note that, regardless of optical thickness and refractive index,
the /3-varying source function results approach nearly the same
asymptote as /3 increases. As ft becomes very large, the results
represent small Tr behavior (see Refs. 5 and 6).
After obtaining the IB-varying solution, numerical integration [e.g., Eqs. (31)] gives the rr-varying solution. Figures 8
239
REGUIGUI AND DOUGHERTY: TWO-DIMENSIONAL RADIATIVE TRANSFER
3.00
2.0,
2.0,
5.0,
5.0,
5.0,
2.0,
n
n
n
n
n
n
= 1.33
= 1.33
= 1.33
= 1.33
= 1.0d
= 1.00
2 0.60
0.00
.
10-3
1Q-2 1Q-1
1.0
101
10)2
103
Fig. 7 Effects of refractive index and optical thickness on the ftvarying intensity for o> = 1.0. (Curves 5 and 6 are from Ref. 16.)
10-3
1
2
3
4
10-4
n
n
n
n
= 1.00, T0 = 2.0
= 1.33, T0 = 2.0
= 1.00, r0 = $.0
= 1.33, T0 = 5.0
X 10-5
o
b
o 1(T
^
and 9 are plots of the reflected intensity outside the medium.
The first plot (Fig. 8) provides a close look at15the trends over
the normal experimental measurement range. Figure 9 gives
the intensity over a wider range, specifically within the original
incident laser beam. The trends clearly show a dramatic increase in the reflected intensity within the original laser beam—which is to be expected. Curves 2 and 4 are for the reflected
intensity from a one-layer medium with top and bottom reflecting boundaries (ri = 1.33). These curves show that the
back reflected intensity increases in value as the optical thickness of the medium increases. This is due to the increased
amount of scattering [events]* Some results for unit refractive
index16 are also included in Fig. 9 for comparison. It is apparent from the plots that assuming unit refractive index in
the case of a water medium would lead to severe overestimatioh of the reflected intensity. However, the general trends
of the two sets of solutions are quite similar.
Conclusions
The theoretical work presented herein will provide a more
flexible and general tool to study a wider range in indices of
refraction, and to investigate their effect on the back-scattered
intensity, than is available in the existing models. The computerized model has the capability of varying the refractive
index, along with handling internal reflection at the boundaries and allowing the albedo to vary through layers of variable optical thickness of the medium being analyzed. With
such a model, validated by experimental
work (now being
conducted at Oklahoma State University15), radiative transfer
within simulated solar ponds or any other multilayered medium can be studied for a much wider variety of parameters
than those assessed by the hardware.
Appendix: Auxiliary Functions
In this appendix, several functions that were referenced in
the body of this paper are written explicitly. In the following
equations, N represents the number of layers, and i represents
a particular layer:
7
& 10~
10,-8
^
10-9
100.0
10.0
1000.0
(Al)
10000.0
(A2)
r/r0
Fig. 8 Effect of refractive index and optical thickness on reflected
intensity far from the incident beam for <D = 1.0. O-Varying results
of Ref. 16 were used to obtain curves 1 and 3.)
)
= [1 - p,,..^^
(A3)
(A4
is defined in the following equations:
1 n = 1.00, T0 = 2.0
2 n = 1.33, T0 = 2.0 3 n = 1.00, TO = 5.0
i) = n[exp[-AV/i,]], for abs(/ - i) > 1 (A5)
4 n = 1.33, TO = 5.0 -
where v - minimum (/,;') + 1, and w = maximum (/,;) 1, and £,(/*./) = 1, for abs(/ — 1)^1. The <l>'s are defined
as follows:
], i >J
(A6)
*=7
where M, = [1 - (1 - M?>4]"2. I f / = A then OJG*,) = 1
10
10-10
0.01
0.1
1.0
10.0
...._______
100.0 1000.0 10000.0
r/r0
Fig. 9 Effect of refractive index and optical thickness on reflected
intensity for <w = 1.0. (0-Varying results of Ref. 16 were used to obtain
curves 1 and 3.)
*S(M») = fl I4JM*K.*-ifaj],i</
(A7)
*-!+!
If i = /, the Q^GI/)
=
1- The definitions for $ are
(A8)
REGUIGUI AND DOUGHERTY: TWO-DIMENSIONAL RADIATIVE TRANSFER
240
where A: = (i 4- ;)/2, m = (3; - 0/2, / = (3i - y)/2. And
for abs(/ - /) = 3, Q and # are given as
(A9)
Z is defined as
X tkj(fJLk)Gjm(lJLj)
(A10)
X
x exp(-2Ar//x/)[p/n(/A/) +
(All)
x C
(A25)
+
, and T, are given by
x
(A12)
x
(A26)
(A13)
for (i + 2) < f
(A14)
where A: = (2; + i)/3, / - (/ + 20/3, m = (4; - 0/3, and r
= (4/ - y)/3. For all of the above definitions of Qnj and Kv,
the IJL values are related as follows: JJL£ = [1 - (1 jit?)n?e]1/2, where e = y, k, /, m, n, or r.
In addition, the G,y values are
If i > N - 2, then 77(/O = 1
*?(/**) = 1 - t2
Ci*(fO, ^r i > 2
=2
Gfa) = ^Mi)p«OA/X^-XM/)exp(-2AT//il)
(A15)
If i < 2, then 77(ju,,) = 1. For F, when j < /
if M/ < ^,2/-y,cr, then Gifa) = 0.
Lastly, for cases where the number of layers is greater than
one, the equation for f'r [Eq. (4)] should be modified to include passage through all intermediate layers as follows:
= (rr)2 + (raiy)2 -
(A16)
When j > ;
(A28)
where
max(/,y)
(A29)
k = min(/,y)
and when / = ;', then
= [!-(!- /Lt/2)n?J1/2. Note that when k = min (/,/),
zk-i — Tz> an<^ when k = max (/,/), r2A. = T^. In the special
case where i = ;, Eq. (A29) becomes Eq. (5).
r,,(M,',T2,T2) = RHjp
T
x exp[-(T2,_, -
(A18)
x exp[-(Tz( - rz
For the B values
r
(A27)
S/T;,T2)*;-2(M/,T2) dr2//ty.
(A19)
./TZ/-1
Acknowledgments
This work was supported in part by the University Center
for Energy Research at Oklahoma State University, Grants
1150714 and 1150718, and by National Science Foundation
Grant CTS 8907149. In addition, the authors are grateful to
IBM and its Palo Alto Scientific Center for a grant of supercomputing time through the Research Support Program.
References
2
f" sw(T;)v;- Ovr;) dry/*,
JT,/-I
(A20)
The two functions Q and ^ are given only for four layers.
For abs(/ - 1) = 1, Q and K are given as
(A21)
(A22)
For abs(/ - /) = 2, 0 and /C are given as
(A23)
(A24)
^uckius, R. O., and Tseng, M. M., "Radiation Heat Transfer in
a Planar Medium with Anisotropic Scattering and Directional Boundaries," Journal of Quantitative Spectroscopy & Radiative Transfer,
Vol. 20, 1978, pp. 385-402.
2
Cengel, Y. A., and Ozisik, M. N., "Radiation Transfer in an
Anisotropically Scattering Slab with Directional Dependent Reflectivities," American Society of Mechanical Engineers Paper 86-HT28, 1986.
3
Armaly, B. F., and Lam, T. T., "Influence of Refractive Index
on Reflectance from a Semi-Infinite Absorbing-Scattering Medium
with Collimated Incident Radiation," International Journal of Heat
and Mass Transfer, Vol. 18, 1975, pp. 893-900.
4
Dougherty, R. L., "Numerical Results for Radiative Transfer in
a Semi-Infinite Absorbing/Scattering Slab Exhibiting Fresnel Reflection," AIAA Paper 88-0076, AIAA 26th Aerospace Sciences Meeting, Reno, NV, 1988.
5
Crosbie, A. L., and Dougherty, R. L., "Two-Dimensional Isotropic Scattering in a Finite Thick Cylindrical Medium Exposed to a
Laser Beam," Journal of Quantitative Spectroscopy & Radiative
Transfer, Vol. 27, No. 2, 1982, pp. 149-183.
6
Crosbie, A. L., and Dougherty, R, L., "Two-Dimensional Ra-
REGUIGUI AND DOUGHERTY; TWO-DIMENSIONAL RADIATIVE TRANSFER
diative Transfer in a Cylindrical Geometry with Anisotrppic Scattering," Journal of Quantitative Spectroscopy & Radiative Transfer, Vol.
25, 1980, pp. 551-569.
7
Stamnes, K., and Conklin, P., "A New Multi Layer Discrete
Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmospheres," Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 31, 1984, pp. 273-282.
"Suttori, W. H., and Kamath, R., "Participating Radiative Heat
Transfer in a Three-Dimensional Rectangular Medium with Layered
Properties," American Society of Mechanical Engineers Paper 86HT-25, 1986.
9
Crosbie, A. L., and Dougherty, R. L., "Influence of Refractive
Index on the Two-Dimensional Back-Scattering of a Laser Beam:
Asymptotic Solutions," Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 40, No. 2, 1988, pp. 123-129.
>°Nelson, H. F., Look, D. C., Jr., and Crosbie, A. L., "TwoDimensional Radiative Back-Scattering from Optically Thick Media," Journal of Heat Transfer, Vol. 108, 1986, pp. 619-625.
"Shouman, S. M., and Ozisik, M, N., "Radiation Transfer in an
Isotropically Scattering Two-Region Slab with Reflecting Boundaries," Journal of Quantitative Spectroscopy & Radiative Transfer, Vol.
26, 1981, pp. 1-9.
12
Reguigui, N. M., "Radiative Transfer in a Four-Layer Cylindrical
Medium with Reflecting Boundaries," Master of Science Thesis,
Oklahoma State Univ., 1990.
13
Reguigui, N. M., and Dougherty, R. L., "Two-Dimensional Radiative Transfer in a Cylindrical Layered Medium with Reflective
Boundaries," AIAA Paper 90-1779, AIAA/ASME 5th Joint Thermophysics and Heat Transfer Conf., Seattle, WA, 1990.
14
Dougherty, R. L., private communication, 1990.
15
Dorri-Nowkoorani, F., Reguigui, N. M., and Dougherty, R. L.,
"Back-Scattering of a Laser Beam from a Layered Cylindrical Medium: Refractive Index Effects," AIAA Paper 90-1764, AIAA/ASME
5th Joint Thermophysics and Heat Transfer Conf., Seattle, WA, 1990.
16
Crosbie, A. L., and Koewing, J. W., "Two-Dimensional Radiative Transfer in a Finite Scattering Planar Medium," Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 21, 1979, pp. 573595.
Recommended Reading from the AIAA
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impact of atmospheric uncertainties and viscous interaction effects), thermal protection, and surface effects such as temperature-dependent reaction rate expressions for
oxygen recombination; surface-ship equations for low-Reynolds-number multicomponent air flow, rate chemistry in flight regimes, and noncatalytic surfaces for metallic
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