The necessity of conserving both scattered energy and asymmetry factor for ballistic incidence after finite volume method (FVM) or discrete-ordinates method (DOM) discretization is shown. A phase-function normalization technique introduced previously by the present authors is applied to scattering of ballistic incidence in 3D FVM/DOM to improve treatment of anisotropic scattering through reduction of angular false scattering errors. Ultrafast radiative transfer predictions generated using FVM and DOM are compared to benchmark Monte Carlo to illustrate the necessity of ballistic phase-function normalization. Proper ballistic phase-function treatment greatly improves predicted heat fluxes and energy deposition for anisotropic scattering and for situations where accurate numerical modeling is crucial.

References

References
1.
Kumar
,
S.
, and
Mitra
,
K.
,
1999
, “
Microscale Aspects of Thermal Radiation Transport and Laser Applications
,”
Adv. Heat Transfer
,
33
, pp.
187
294
.10.1016/S0065-2717(08)70305-8
2.
Guo
,
Z.
, and
Hunter
,
B.
,
2013
, “
Advances in Ultrafast Radiative Transfer Modeling and Applications: A Review
,”
Heat Transfer Res.
,
44
(
3–4
), pp.
303
344
.10.1615/HeatTransRes.2012006387
3.
Qiu
,
T. Q.
, and
Tien
,
C. L.
,
1993
, “
Heat Transfer Mechanisms During Short-Pulse Laser Heating of Metals
,”
ASME J. Heat Transfer
,
115
(
4
), pp.
835
841
.10.1115/1.2911377
4.
Choi
,
T. Y.
, and
Grigoropoulos
,
C. P.
,
2002
, “
Plasma and Ablation Dynamics in Ultrafast Laser Processing of Crystalline Silicon
,”
J. Appl. Phys.
,
92
(9), pp.
4918
4925
.10.1063/1.1510565
5.
Chimmalgi
,
A.
,
Choi
,
T. Y.
,
Grigoropoulos
,
C. P.
, and
Komvopoulos
,
K.
,
2003
, “
Femtosecond Laser Apertureless Near-Field Nanomachining of Metals Assisted by Scanning Probe Microscopy
,”
Appl. Phys. Lett.
,
82
(
8
), pp.
1146
1148
.10.1063/1.1555693
6.
Cheng
,
C.
, and
Xu
,
X.
,
2005
, “
Mechanisms of Decomposition of Metal During Femtosecond Laser Ablation
,”
Phys. Rev. B
,
72
(
16
), pp.
5415
5429
.10.1103/PhysRevB.72.165415
7.
Wang
,
L.
,
Ho
,
P. P.
,
Liu
,
C.
,
Zhang
,
G.
, and
Alfano
,
R. R.
,
1991
, “
Ballistic 2-D Imaging Through Scattering Walls Using an Ultrafast Optical Kerr Gate
,”
Science
,
253
(
5021
), pp.
769
771
.10.1126/science.253.5021.769
8.
Yoo
,
K. M.
,
Liu
,
F.
, and
Alfano
,
R. R.
,
1993
, “
Ultrafast Time-Gated Imaging in Thick Tissues: A Step Toward Optical Mammography
,”
Opt. Lett.
,
18
(
13
), pp.
1092
1094
.10.1364/OL.18.001092
9.
Yodh
,
A.
, and
Chance
,
B.
,
1995
, “
Spectroscopy and Imaging With Diffuse Light
,”
Phys. Today
,
48
(
3
), pp.
34
40
.10.1063/1.881445
10.
Niemz
,
M. H.
,
Klancnik
,
E. G.
, and
Bille
,
J. F.
,
1991
, “
Plasma-Mediated Ablation of Corneal Tissue at 1053 nm Using a Nd:YLF Oscillator/Regenerative Amplifier Laser
,”
Lasers Surg. Med.
,
11
(
5
), pp.
426
431
.10.1002/lsm.1900110507
11.
Kim
,
B.-M.
,
Feit
,
M. D.
,
Rubenchik
,
A. M.
,
Joslin
,
E. M.
,
Celliers
,
P. M.
,
Eichler
,
J.
, and
Da Silva
,
L. B.
,
2001
, “
Influence of Pulse Duration on Ultrashort Laser Pulse Ablation of Biological Tissues
,”
J. Biomed. Opt.
,
6
(
3
), pp.
332
338
.10.1117/1.1381561
12.
Huang
,
H.
, and
Guo
,
Z.
,
2010
, “
Ultrashort Pulsed Laser Ablation and Stripping of Freeze-Dried Dermis
,”
Lasers Med. Sci.
,
25
(
4
), pp.
517
524
.10.1007/s10103-009-0741-9
13.
Guo
,
Z.
,
Wang
,
X. L.
, and
Huan
,
H.
,
2010
, “
Plasma-Mediated Ablation of Biofilm Contamination
,”
Appl. Surf. Sci.
,
257
(
4
), pp.
1247
1253
.10.1016/j.apsusc.2010.08.033
14.
Jaunich
,
M.
,
Raje
,
S.
,
Kim
,
K. H.
,
Mitra
,
K.
, and
Guo
,
Z.
,
2008
, “
Bio-Heat Transfer Analysis During Short Pulse Irradiation of Tissues
,”
Int. J. Heat Mass Transfer
,
51
(
23–24
), pp.
5511
5521
.10.1016/j.ijheatmasstransfer.2008.04.033
15.
Jiao
,
J.
, and
Guo
,
Z.
,
2011
, “
Modeling of Ultrashort Pulsed Laser Ablation in Water and Biological Tissues in Cylindrical Coordinates
,”
Appl. Phys. B
,
103
(
1
), pp.
195
205
.10.1007/s00340-010-4197-1
16.
Bass
,
L. S.
, and
Treat
,
M. R.
,
1995
, “
Laser Tissue Welding: A Comprehensive Review of Current and Future Applications
,”
Lasers Surg. Med.
,
17
(
4
), pp.
315
349
.10.1002/lsm.1900170402
17.
Kim
,
K. H.
, and
Guo
,
Z.
,
2004
, “
Ultrafast Radiation Heat Transfer in Laser Tissue Welding and Soldering
,”
Numer. Heat Transfer, Part A
,
46
(
1
), pp.
23
40
.10.1080/10407780490457365
18.
Guo
,
Z.
, and
Kumar
,
S.
,
2001
, “
Discrete-Ordinates Solution of Short-Pulsed Laser Transport in Two-Dimensional Turbid Media
,”
Appl. Opt.
,
40
(
19
), pp.
3156
3163
.10.1364/AO.40.003156
19.
Mitra
,
K.
, and
Kumar
,
S.
,
1999
, “
Development and Comparison of Models for Light-Pulse Transport Through Scattering-Absorbing Media
,”
Appl. Opt.
,
38
(
1
), pp.
188
196
.10.1364/AO.38.000188
20.
Guo
,
Z.
, and
Kumar
,
S.
,
2002
, “
Three-Dimensional Discrete Ordinates-Method in Transient Radiative Transfer
,”
J. Thermophys. Heat Transfer
,
16
(
3
), pp.
289
296
.10.2514/2.6689
21.
Sakami
,
M.
,
Mitra
,
K.
, and
Hsu
,
P.
,
2002
, “
Analysis of Light-Pulse Transport Through Two-Dimensional Scattering-Absorbing Media
,”
J. Quant. Spectrosc. Radiat. Transfer
,
73
(
2–5
), pp.
169
179
.10.1016/S0022-4073(01)00216-3
22.
Das
,
C.
,
Trivedi
,
A.
,
Mitra
,
K.
, and
Vo-Dinh
,
T.
,
2003
, “
Experimental and Numerical Analysis of Short Pulse Laser Interaction With Tissue Phantoms Containing Inhomogeneities
,”
Appl. Opt.
,
42
(
25
), pp.
5173
5180
.10.1364/AO.42.005173
23.
Akamatsu
,
M.
, and
Guo
,
Z.
,
2011
, “
Ultrafast Radiative Heat Transfer in Three-Dimensional Highly-Scattering Media Subjected to Pulse Train Irradiation
,”
Numer. Heat Transfer, Part A
,
59
(
9
), pp.
653
671
.10.1080/10407782.2011.572751
24.
Chai
,
J. C.
,
Lee
,
H. S.
, and
Patankar
,
S. V.
,
1993
, “
Ray Effect and False Scattering in the Discrete Ordinates Method
,”
Numer. Heat Transfer, Part B
,
24
(
4
), pp.
373
389
.10.1080/10407799308955899
25.
Hunter
,
B.
, and
Guo
,
Z.
,
2012
, “
Conservation of Asymmetry Factor in Phase Function Discretization for Radiative Transfer Analysis in Anisotropic Scattering Media
,”
Int. J. Heat Mass Transfer
,
55
(
5–6
), pp.
1544
1552
.10.1016/j.ijheatmasstransfer.2011.11.009
26.
Hunter
,
B.
, and
Guo
,
Z.
,
2012
, “
Reduction of Angle Splitting and Computational Time for the Finite Volume Method Via Phase Function Normalization
,”
Int. J. Heat Mass Transfer
,
55
(
9–10
), pp.
2449
2460
.10.1016/j.ijheatmasstransfer.2012.01.015
27.
Chai
,
J. C.
,
2003
, “
One-Dimensional Transient Radiation Heat Transfer Modeling Using a Finite-Volume Method
,”
Numer. Heat Transfer, Part B
,
44
(
2
), pp.
187
208
.10.1080/713836346
28.
Chai
,
J. C.
,
2004
, “
Transient Radiative Transfer in Irregular Two-Dimensional Geometries
,”
J. Quant. Spectrosc. Radiat. Transfer
,
84
(
3
), pp.
281
294
.10.1016/S0022-4073(03)00183-3
29.
Chai
,
J. C.
,
Hsu
,
P.-F.
, and
Lam
,
Y. C.
,
2004
, “
Three-Dimensional Transient Radiative Transfer Modeling Using the Finite Volume Method
,”
J. Quant. Spectrosc. Radiat. Transfer
,
86
(
3
), pp.
299
313
.10.1016/j.jqsrt.2003.08.008
30.
Muthukumaran
,
R.
, and
Mishra
,
S. C.
,
2008
, “
Transient Response of a Planar Participating Medium Subjected to a Train of Short-Pulse Irradiation
,”
Int. J. Heat Mass Transfer
,
51
(
9
), pp.
2418
2432
.10.1016/j.ijheatmasstransfer.2007.08.035
31.
Muthukumaran
,
R.
, and
Mishra
,
S. C.
,
2008
, “
Thermal Signatures of a Localized Inhomogeneity in a 2-D Participating Medium Subjected to an Ultra-Fast Step-Pulse Laser Wave
,”
J. Quant. Spectrosc. Radiat. Transfer
,
109
(
5
), pp.
705
726
.10.1016/j.jqsrt.2007.08.025
32.
Rahmani
,
R. K.
,
Ayasoufi
,
A.
, and
Molavi
,
H.
,
2009
, “
Numerical Simulation of Transient Radiative Heat Transfer Applying Finite Volume Method on Generalized Computational Grid
,”
Proceedings of the ASME Early Career Technical Conference
, ASME, New York, pp. 33–39.
33.
Mishra
,
S. C.
,
Chugh
,
P.
,
Kumar
,
P.
, and
Mitra
,
K.
,
2006
, “
Development and Comparison of the DTM, the DOM and the FVM Formulations for the Short-Pulse Laser Transport Through a Participating Medium
,”
Int. J. Heat Mass Transfer
,
49
(
11–12
), pp.
1820
1832
.10.1016/j.ijheatmasstransfer.2005.10.043
34.
Kim
,
T. K.
, and
Lee
,
H.
,
1988
, “
Effect of Anisotropic Scattering on Radiative Heat Transfer in Two-Dimensional Rectangular Enclosures
,”
Int. J. Heat Mass Transfer
,
31
(
8
), pp.
1711
1721
.10.1016/0017-9310(88)90283-9
35.
Boulet
,
P.
,
Collin
,
A.
, and
Consalvi
,
J. L.
,
2007
, “
On the Finite Volume Method and the Discrete Ordinates Method Regarding Radiative Heat Transfer in Acute Forward Anisotropic Scattering Media
,”
J. Quant. Spectrosc. Radiat. Transfer
,
104
(
3
), pp.
460
473
.10.1016/j.jqsrt.2006.09.010
36.
Hunter
,
B.
, and
Guo
,
Z.
,
2012
, “
Phase Function Normalization for Accurate Analysis of Ultrafast Collimated Radiative Transfer
,”
Appl. Opt.
,
51
(
12
), pp.
2192
2201
.10.1364/AO.51.002192
37.
Hunter
,
B.
, and
Guo
,
Z.
,
2012
, “
Phase-Function Normalization in 3-D Discrete-Ordinates Solution of Radiative Transfer—Part I: Conservation of Scattered Energy and Asymmetry Factor
,”
Numer. Heat Transfer, Part B
,
62
(
4
), pp.
203
222
.10.1080/10407790.2012.709163
38.
Hunter
,
B.
, and
Guo
,
Z.
,
2012
, “
Phase-Function Normalization in 3-D Discrete-Ordinates Solution of Radiative Transfer—Part II: Benchmark Comparisons
,”
Numer. Heat Transfer, Part B
,
62
(
4
), pp.
223
242
.10.1080/10407790.2012.709165
39.
Chai
,
J. C.
,
Lee
,
H. S.
, and
Patankar
,
S. V.
,
1994
, “
Finite Volume Method for Radiation Heat Transfer
,”
J. Thermophys. Heat Transfer
,
8
(
3
), pp.
419
425
.10.2514/3.559
40.
Murthy
,
J. Y.
, and
Mathur
,
S. R.
,
1998
, “
Finite Volume Method for Radiative Heat Transfer Using Unstructured Meshes
,”
J. Thermophys. Heat Transfer
,
12
(
3
), pp.
313
321
.10.2514/2.6363
41.
Kim
,
S. H.
, and
Huh
,
K. Y.
,
2000
, “
A New Angular Discretization Scheme of the Finite Volume Method for 3-D Radiative Heat Transfer in Absorbing, Emitting, and Anisotropically Scattering Media
,”
Int. J. Heat Mass Transfer
,
43
(
7
), pp.
1233
1242
.10.1016/S0017-9310(99)00211-2
42.
Collin
,
A.
,
Consalvi
,
J. L.
, and
Boulet
,
P.
,
2011
, “
Acute Anisotropic Scattering in a Medium Under Collimated Irradiation
,”
Int. J. Therm. Sci.
,
50
(
1
), pp.
19
24
.10.1016/j.ijthermalsci.2010.09.002
43.
Chui
,
E. H.
,
Raithby
,
G. D.
, and
Hughes
,
P. M. J.
,
1992
, “
Prediction of Radiative Transfer in Cylindrical Enclosures With the Finite Volume Method
,”
J. Thermophys. Heat Transfer
,
6
(
4
), pp.
605
611
.10.2514/3.11540
44.
Longoni
,
G.
, and
Haghighat
,
A.
,
2001
, “
Development of New Quadrature Sets With the “Ordinate Splitting” Technique
,”
Proceedings of the ANS International Meeting on Mathematical Methods for Nuclear Applications
, Salt Lake City, UT.
45.
Lee
,
H.
, and
Buckius
,
R. O.
,
1982
, “
Scaling Anisotropic Scattering in Radiation Heat Transfer for a Planar Medium
,”
ASME J. Heat Transfer
,
104
(
1
), pp.
68
75
.10.1115/1.3245070
46.
Guo
,
Z.
, and
Kumar
,
S.
,
2000
, “
Equivalent Isotropic Scattering Formulation for Transient Short-Pulse Radiative Transfer in Anisotropic Scattering Planar Media
,”
Appl. Opt.
,
39
(
24
), pp.
4411
4417
.10.1364/AO.39.004411
47.
Hunter
,
B.
, and
Guo
,
Z.
,
2011
, “
Comparison of Discrete-Ordinates Method and Finite Volume Method for Steady-State and Ultrafast Radiative Transfer Analysis in Cylindrical Coordinates
,”
Numer. Heat Transfer, Part B
,
59
(
5
), pp.
339
359
.10.1080/10407790.2011.572719
48.
Cheong
,
W. F.
,
Prahl
,
S. A.
, and
Welch
,
A. J.
,
1990
, “
A Review of the Optical Properties of Biological Tissues
,”
IEEE J. Quantum Electron.
,
26
(12), pp.
2166
2185
.10.1109/3.64354
49.
Hunter
,
B.
, and
Guo
,
Z.
,
2015
, “
Numerical Smearing, Ray Effect, and Angular False Scattering in Radiation Transfer Computation
,”
Int. J. Heat Mass Transfer
,
81
, pp.
63
74
.10.1016/j.ijheatmasstransfer.2014.10.014
You do not currently have access to this content.