Track 1 TOC
Proceedings
Proceedings
of HT2003 of
2003ASME
ASMESummer
Summer
Heat
Transfer
Conference
Heat Transfer Conference
July
21–23,
2003,
Las
Vegas,
Nevada,
USA
July
21-23,
2003,
Las
Vegas,
Nevada,
USA
HT2003-47455
HT2003-47455
TRANSIENT RADIATIVE TRANSFER OF LIGHT PULSE PROPAGATION IN THREEDIMENSIONAL SCATTERING MEDIA WITH FINITE VOLUME METHOD AND INTEGRAL
EQUATION MODEL
Xiaodong Lu and Pei-feng Hsu
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
Melbourne, Florida 32901, U.S.A.
and
John C. Chai
School of Mechanical and Production Engineering
Nanyang Technological University
Singapore 639798
INTRODUCTION
The transient radiative transfer process is studied with a finite
volume method (FVM) and an integral equation (IE) model.
Propagation of a short light pulse in the three-dimensional
absorbing and isotropic scattering media is considered.
Collimated irradiation enters at one side of the rectangular
medium. The other five boundaries are cold and black, nonparticipating surfaces. The spatial and temporal distributions of
the integrated intensity and radiative flux are obtained.
thick media. Furthermore, these methods can't resolve the
wave front and miss important physical detailed information in
the early stage of the transient process. It is therefore essential
and critical to develop accurate solution models to advance the
application of transient radiation transport, especially the use of
ultra-short pulsed lasers for optical imaging and remote
sensing. In this study, the results from two different solution
methods are compared and the effect of scattering albedo on a
base case is examined using the integral equation model.
There are many existing models that have been used to
treat transient radiation processes: from the earlier work of
Monte Carlo method to more recently the discrete ordinates,
finite volume, and integral equation methods. Most existing
transient models either failed to address the hyperbolic wave
nature of the process or relied on the approximations that
simplify the transport equation, e.g., diffusion approximation
and spherical harmonics approximation, and the geometry
under consideration was usually a one-dimensional slab except
a few cases, where two- and three-dimensional rectangular
geometries were studied. An evident drawback of the these
solution methods is the failure to predict the correct
propagation speed within the medium as well as inadequate
solution accuracy except under certain conditions like optically
NOMENCLATURES
a
coefficient of the FVM discretization equation
b
source term of the FVM discretization equation
c
propagation speed of radiation transport in the
medium
l
l
Dcw
, Dce
direction cosine integrated over ∆Ω l
l
l
Dcs
, Dcn
direction cosine integrated over ∆Ω l
l
l
Dcb
, Dct
direction cosine integrated over ∆Ω l
G
incident radiation, integrated intensity, or fluence
rate
I
radiation intensity
1
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kˆ
unit vector in the z-direction
radiative flux
geometric path length
sˆ
unit vector along a given direction
t
time
u
unit step function
x, y, z
rectangular coordinates
M
total number of control angles
±
±
S cb
, S pb additional source terms for high-resolution scheme
± S±
S ce
, pe additional source terms for high-resolution scheme
± S±
S cn , pn additional source terms for high-resolution scheme
±
±
S cs
, S ps additional source terms for high-resolution scheme
S ct± , S ±pt additional source terms for high-resolution scheme
±
S cw
, S ±pw additional source terms for high-resolution scheme
l
Sm
modified source function
q
s
TRANSIENT RADIATION TRANSPORT PROCESS
The transient radiative transfer equation (TRTE) in an
absorbing, non-emitting, and scattering medium in direction sˆ
can be written as (Ozisik, 1973)
∂I (x, y , z , sˆ, t ) ∂I (x, y , z , sˆ, t )
+
= − β I (x, y , z , sˆ, t )
c∂t
∂s
+
σ
I (x, y , z , sˆ ′, t ) Φ(sˆ ′, sˆ ) dΩ
∫
4π Ω =4π
(1)
FINITE-VOLUME METHOD
The TRTE for a discrete direction l can be written as
∂I l ∂I l
l
+
= − β ml I l + S m
∂s
c∂t
(2)
where the modified extinction coefficient β ml and the modified
l
source function S m
are (Chai et al., 1994)
Greek symbols
β
extinction coefficient
κ
absorption coefficient
σ
scattering coefficient
τ
optical thickness of the medium, β zo
θ
polar angle
ϕ
azimuthal or circumferential angle
Φ
scattering phase function
Ω
solid angle
ω
scattering albedo
β ml = κ + σ −
l
= κI b +
Sm
σ
4π
σ
Φ ll ∆Ω l
(3a)
l ′ l ′l
l′
∑ I Φ ∆Ω
(3b)
4π
M
l ′=1,l ′≠l
Following the procedure outlined in Chai (2003), the final
discretization equation for a typical three-dimensional control
volume (Fig. 1a) and control angle (Fig. 1b) can be written as
l l
a Pl I Pl = aW
IW + a El I El + a Sl I Sl
Subscripts
B
blackbody
C
collimated component or direction
E, W, N, S, B, T east, west neighbors of P
e, w,n, s,b, t
east, west control-volume faces
WW
west neighbor of W
P
control volume
p
pulse temporal width
v
quantity related to the volume element
w
quantity related to the surface (or wall) element
l
+ a lN I N
+ a Bl I Bl + aTl I Tl + b l
(4a)
where
l
a El = − Ae Dce
l
l
aW
= − Aw D cw
(4b)
l
a lN = − An Dcn
l
a Sl = − As Dcs
(4c)
l
aTl = − At Dct
l
a Bl = − Ab Dcb
(4d)
l
l
l
a Pl = Ae Dce
+ Aw Dcw
+ An Dcn
l
l
l
+ As Dcs
+ Ab Dcb
+ At Dct
Superscripts
'
dummy variables
^
vector
o
previous time step
l
angular directions
+ β ml , p ∆V P ∆Ω l +
l
l
bl = Sm
,P ∆V P ∆Ω +
2
(4e)
∆v∆Ω l
+ S c,CLAM
c
∆v∆Ω l o
I P + S p ,CLAM
c
(4f)
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perpendicular to the z = 0 wall.
participating radioactively.
In Eq. (4), SCCLAM and Sp,CLAM are the extra source terms due to
the CLAM scheme.
The time evolution of the incident radiation at five spatial
locations namely, z/L = 1/34, 9/34, 1/2, 25/34, and 33/34 are
plotted in Fig. 2. Results obtained using the IE and FV methods
are shown. The step and CLAM schemes are used with the FV
method. Overall, the IE method is more accurate than the FV
method. Due to false scattering, the FV method solutions
spread over a larger time interval as time increases. As
expected, the step scheme produces more spreading than the
CLAM scheme. This leads to lower peak incident intensities at
the various spatial locations. At a given location, it is also
noticed that radiation penetrates much faster and attenuates
much slower with the step scheme. The solutions obtained
using the CLAM scheme are slightly better than those of the
step scheme. However, it also predicted lower peak incident
radiation, faster penetration time and slower decay time than
the IE formulation.
INTEGRAL EQUATION MODEL
Following the procedure described in Tan and Hsu (2001;
2002), the integrated intensity is
G ( x, y , z , t )
= ∫4π I (x w , y w ,0, sˆ, t − s / c )e − βs dΩ
[
]
1
s
− β ( s − s ′) ′
ds dΩ
∫ ∫ σ ( x ′, y ′, z ′)G (x ′, y ′, z ′, t 2′ )e
4π 4π 0
I (x w , y w ,0, sˆ, t − s / c )e − βs
= ∫∫A(t )
cosθdA
(5)
s2
+
+
1
σ (x ′, y ′, z ′)G ( x ′, y ′, z ′, t ′)e − β ( s − s )
dV
∫∫∫V (t )
4π
( s − s ′) 2
′
The boundary condition is a cube with one black and diffusely
emitting surface at z = 0 and the other five surfaces are cold and
black (or transparent). For a collimated pulse with the intensity
I0 is suddenly emitted from boundary at z = 0 at the instant of t
= 0 and turned off at t = tp, then it can be expressed as
[
(
)]
I (x w , y w ,0, sˆ, t ) = I 0 u (t ) − u t − t p kˆ
An improvement of the FVM result can readily be made is
by decomposing the intensity into the collimated and diffuse
components (Sakami, et al. 2000),
I ( x, y, z , sˆ, t ) = I c ( x, y, z , sˆc , t ) + I d ( x, y, z , sˆ, t )
(6)
 
z 
z
z
1 

G (x, y, z , t ) = I 0 u t −  − u t − − t p  e −κz +
u t −  ⋅
4π  c 

  c  c
∫∫∫
− β ( x − xv )2 +( y − yv )2 +( z − z ′)2
( x − x v ) 2 + ( y − y v ) 2 + ( z − z ′) 2
(8)
In this equation, Ic is the residual intensity of the collimated
beam after extinction, and Id is the diffuse intensity, which is
the result of the scattered radiation away from the collimated
beam.
where k̂ is the unit vector along the collimated beam direction,
i.e., z-axis. The resulting equation of G is
σ (x ′, y ′, z ′)G ( x ′, y ′, z ′, t ′)e
The other walls are not
dx v dy v dz ′
(7)
Equation (7) is a Volterra integral equation of the second kind.
The numerical quadrature used in solving this equation is based
on the YIX method (Hsu, et al. 1993). An asymmetric spherical
ring angular quadrature set is used in the spatial integration (Lu
and Hsu, 2003). The integration limit is a time dependent
domain of influence, which is detailed in Tan and Hsu (2002)
and will not be repeated here.
RESULTS AND DISCUSSION
In this article, transient radiative transfer in a cube subjected to
a collimated beam is examined. The cold medium absorbs and
scatters energy isotropically. The optical thickness of the
medium is 1. Three scattering albedoes namely, 0.1, 0.5 and 1
are examined. The collimated beam enters the domain
3
The effect of scattering albedo is shown in figures 3-5. In
Fig. 3, there are several crossovers of the temporal curves at
different locations. This is caused by the drastic change in the
slope of the temporal curves. For example, the G function has
a sudden drop in magnitude near t/∆t = 35. This corresponds to
the temporal variation in the domain of influence, i.e., abrupt
change in the volume integration limit of Eq. (7). Similar
temporal changes are also observed at other locations. In fact,
the crossover occurs several times over the duration of the
simulation. Same crossover is also seen in the other two cases
(Figs. 4 and 5). However, the effect becomes weaker as albedo
increases. This is expected as the volume integration term, the
second term on the right hand side of Eq. (7), become more
important as albedo increases. The higher signal level in large
albedo case tends to smear out the crossover. In all cases, the
multiple, diffuse scattering effect is not very significant because
the optical thickness is relatively small. For a typical pulse
laser probed medium, optical thickness in the order of 100 and
albedo close to 1 is very common (Hsu, 2000). In that case,
stronger diffuse scattering will result a second, stronger signal
level peak observed in the experiments and our earlier
numerical work.
3
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CONCLUSION
Among several transient radiation transport processes, we
believe the collimated pulse irradiation problem presents a
good test of the numerical models as well as close relevance to
actual physical applications. A high order upwind scheme that
can capture and resolve wave front is essential. This study
compares two important methods: finite volume method and
integral equation model. The CLAM scheme is more accurate
than the simple step scheme used in finite volume method.
However, both schemes are unable to recover the correct
propagation speed as is the integral equation model. The effect
of scattering is not very significant in the case of unity optical
thickness of the medium. The effect will be more significant in
larger optical thickness media. Future work will include the
temporal reflectance and transmittance signals of pulse
irradiated nonhomogeneous media.
Hsu, P.-f. (2001), "Effects of Multiple Scattering and Reflective
Boundary on the Transient Radiative Transfer Process," Int.
J. of Thermal Sciences, Vol. 40, No. 6, pp. 539-549.
Hsu, P.-f., Z. Tan and J. R. Howell (1993), "Radiative Transfer
by the YIX Method in Nonhomogeneous, Scattering and
Non-Gray Medium," AIAA J. Thermophysics and Heat
Transfer Vol. 7, No. 3, pp.487-495.
Lu, X. and P.-f. Hsu (2003), "Parallel Computing Performance
of Two Numerical Quadratures for an Integral Formulation
of Transient Radiative Transfer Process," under review in
AIAA J. Thermophysics and Heat Transfer.
Ozisik, M. N. (1973), Radiative Transfer and Interaction with
Conduction and Convection, J. Wiley, New York.
Sakami, M., K. Mitra, and P.-f. Hsu (2000), "Transient
Radiative Transfer in Anisotropically Scattering Media
using Monotonicity-Preserving Schemes," ASME 2000 Int.
Mechanical Engineering Congress & Exposition, ASME
HTD-Vol. 366-1, pp. 135-143, Orlando, FL, November
2000.
Tan, Z.-M., and P.-f. Hsu (2001), "An Integral Formulation of
Transient Radiative Transfer," ASME J. Heat Transfer, Vol.
123, No. 3, pp.466-475.
Tan, Z.-M., and P.-f. Hsu (2002), "Transient Radiative Transfer
in Three-Dimensional Homogeneous and Nonhomogeneous
Participating Media," J. Quant. Spect. & Rad. Transfer, Vol.
73, No. 2-5, pp. 181-194.
References
Chai, J. C. (2003), "One-Dimensional Transient Radiation Heat
Transfer Modeling Using a Finite-Volume Method,"
Numerical Heat Transfer, Part B, Accepted for publication.
Chai, J. C., H. S. Lee, and S. V. Patankar, (1994), "Improved
Treatment of Scattering Using the Discrete Ordinates
Method," ASME J. Heat Transfer, Vol. 116, No. 1, pp. 260263.
4
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N
10
T
n
t
P
E
e
W
B
G(0,0,z,t)
w
s
b
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
z
0
20
G(0,0,z,t)
y
+
φ− φ
x
40
50
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
IE
CLAM
Step
1/2
0.6
G(0,0,z,t)
9/34
25/34
33/34
0.4
0.2
10
t/∆t
15
ω = 0.5
0
10
1
z/L=1/34
z=1/34
z=9/34
z=1/2
z=25/34
z=33/34
10
20
t/∆t
30
40
50
Figure 4: Pulse propagation in three-dimensional media with
scattering albedo = 0.5, based on IE model.
Figure 1: (a) A typical two-dimensional control volume, (b)
a typical control angle.
5
30
0
(b)
0.8
t/∆t
Control angle, ∆Ωl
θ−
θ+
G(0,0,z,t)
10
Figure 3: Pulse propagation in three-dimensional media with
scattering albedo = 0.1, based on IE model.
10
0
ω = 0.1
S
(a)
0
z=1/34
z=9/34
z=1/2
z=25/34
z=33/34
20
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
z=1/34
z=9/34
z=1/2
z=25/34
z=33/34
ω=1
0
10
20
t/∆t
30
40
50
Figure 5: Pulse propagation in three-dimensional media with
scattering albedo = 1, based on IE model.
Figure 2: Comparison of integral equation model and finite
volume method with two difference schemes (CLAM and step).
5
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