One-dimensional transient radiative transfer by lattice Boltzmann method.

One-dimensional transient radiative transfer by lattice Boltzmann method Yong Zhang,1 Hongliang Yi,1,* and Heping Tan2 1

School of Energy Science and Engineering, Harbin Institute of Technology, 92 West Dazhi Street, Harbin 150001, China 2 [email protected] *[email protected]

Abstract: The lattice Boltzmann method (LBM) is extended to solve transient radiative transfer in one-dimensional slab containing scattering media subjected to a collimated short laser irradiation. By using a fully implicit backward differencing scheme to discretize the transient term in the radiative transfer equation, a new type of lattice structure is devised. The accuracy and computational efficiency of this algorithm are examined firstly. Afterwards, effects of the medium properties such as the extinction coefficient, the scattering albedo and the anisotropy factor, and the shapes of laser pulse on time-resolved signals of transmittance and reflectance are investigated. Results of the present method are found to compare very well with the data from the literature. For an oblique incidence, the LBM results in this paper are compared with those by Monte Carlo method generated by ourselves. In addition, transient radiative transfer in a two-Layer inhomogeneous media subjected to a short square pulse irradiation is investigated. At last, the LBM is further extended to study the transient radiative transfer in homogeneous medium with a refractive index discontinuity irradiated by the short pulse laser. Several trends on the timeresolved signals different from those for refractive index of 1 (i.e. refractive-index-matched boundary) are observed and analysed. ©2013 Optical Society of America OCIS codes: (000.3860) Mathematical methods in physics; (010.5620) Radiative transfer; (140.7090) Ultrafast lasers; (290.7050) Turbid media.

References and links 1. 2.

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#194705 - $15.00 USD Received 26 Jul 2013; revised 15 Sep 2013; accepted 30 Sep 2013; published 7 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.024532 | OPTICS EXPRESS 24532

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1. Introduction In the past twenty years, due to the availability of short-pulse lasers, transient radiative transfer (TRT) in participating media has received considerable attention in many emerging applications. The developments in micro/nano-scale systems [1, 2], short-pulsed laser in materials [3], optical tomography [4], laser therapy [5], particle detection and sizing [6] and


other applications have indicated that TRT is an important process which requires rigorous study. A detailed review on various aspects of the transient radiative transfer behavior induced by a short-pulsed laser was presented by Kumar and Mitra [7]. An interaction of a short-pulse radiation with a participating medium results in temporal signals. The time-resolved radiative signals offer some unique characteristics that are not available from the steady-state light sources. Usually the temporal or time-resolved radiative signals (transmittance and/or reflectance) are used to determine the medium’s internal structure or/and radiative properties. Hence, sufficiently accurate and efficient solution methods are required. In the recent decade, with the start of research in the area of transient radiative transfer, several numerical strategies have been developed, including the Monte Carlo method (MCM), discrete ordinate method (DOM), integral equation (IE) models, finite volume method FVM, and the discontinuous finite element method (DFEM). The Monte Carlo method was used to model the transient radiative transfer by Schweiger et al. [8] and Guo et al. [9]. However, the Monte Carlo method requires a large number of energy bundles to obtain accurate and smooth results, and is so computationally expensive. Lu and Hsu [10, 11] developed a reverse Monte Carlo (RMC) method which shortened the computation time and improved the computational efficiency in the investigation of transient radiative transfer. Martinelli et al. [12] presented a theoretical analysis that provided a clear line of derivation from the RTE to the scaling relations which formed the basis of single Monte Carlo (sMC). Guo and Kumar [13] and Sakami et al. [14] extended applications of the DOM to the 2-D rectangular enclosure. Application of the DOM to the 3-D rectangular enclosure was extended by Guo and Kumar [15]. Wu et al. [16] applied the DOM with a firstorder spatial scheme and a modified DOM (MDOM) to solve transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate exposed to a diffuse strong irradiation at one of its boundaries. Tan and Hsu [17] developed an integral equation (IE) formulation to treat the general transient radiative transfer equation, and Wu [18] used a slightly different form of IE formulation to study the light pulse transport in a participating planar medium. Their solutions provided very accurate results and were verified with Monte Carlo algorithms [19]. Chai [20] introduced the FVM to solve the transient radiative transfer in a one-dimensional absorbing and isotropically scattering planar medium. Kim et al. [21] investigated the transient radiative heat transfer in one-dimensional slabs separated by a participating media using the FVM, and in the work the convection schemes in computational fluid dynamics (CFD) such as step, diamond, 2nd order upwind, QUICK, and CLAM were introduced to capture the physics of the radiative wave propagation. Ruan et al. [22] used the FVM to solve the transient radiative transfer problem in 1-D homogeneous and inhomogeneous media irradiated by the short pulse laser. A non-dimensional number was proposed to analyze the characteristics of the temporal transmittance and reflectance signals in [22]. Mishra et al. [23] provided the analysis of transient radiative transfer caused by a short-pulse laser irradiation on a participating media using the discrete transfer method (DTM), DOM, and FVM. Liu and associates [24, 25] extended the application of discontinuous finite element method (DFEM) based on the discrete ordinates equation to solving transient radiative transfer in absorbing, emitting, and scattering media, with the use of a time shift and superposition principle for improving the computational efficiency. The lattice Boltzmann method (LBM) is a relatively new computational tool. In the recent decades, it has emerged as an efficient method to analyze a vast range of problems in fluid flow and heat transfer [26–28]. This surge in applications of the LBM is owing to its attractive properties of simple implementation on the computer, mesoscopic nature, ability to handle complex geometry and boundary conditions, capability of stable and accurate simulation, and the inherent parallel nature. Over the years, the LBM has been applied to solve the energy equations of combined mode conduction and/or convection problems [29–32] involving volumetric radiation in


which the radiative information has been computed using the conventional numerical methods like the DTM [29], FVM [30–32]. Recently, the LBM itself has been adopted for solving radiation transport problems [33–37] in which one-dimensional (1D) and twodimensional (2D) examples of radiative transfer were discussed. Asinari and associates [33, 34] described the advantage of having common data structures for radiation intensity and fluid flow in radiative heat transfer and fluid mechanics problems. They extended the application of the LBM to solve a benchmark radiative equilibrium problem involving a 2-D rectangular enclosure, and the LBM was found to have an edge over the FVM. Based on the Chapman-Enskog method, Ma et al. [35] proposed the lattice Boltzmann model for onedimensional radiative transfer from the Boltzmann equation. Bindra et al. [36] extended the LBM to solving the radiative or neutron transport equation with considering the scattering term. Mishra and associates [37] extended the LBM to analysis of transport of collimated radiation in a participating media. However, to the best of our knowledge, as a promising numerical scheme, the LBM has not been used for the solution to transient radiative transfer in the participating medium irradiated by the short pulse laser. In this article we present a LBM for solving the transient radiative transfer in one-dimensional slab containing absorbing, scattering media subjected to the collimated short laser irradiation. No research works have been carried out to investigate the transient radiative transfer in homogeneous medium with a refractive index discontinuity irradiated by the short pulse laser. Thus, we further extend the application of LBM to studying this problem. The outline of this paper is as below. In the following section, the framework of lattice Boltzmann method for solving the transient radiative transfer is formulated and the solution process about the implementation of LBM is presented. In section 3, the accuracy and efficiency of this algorithm are studied firstly. Afterwards, to show the flexibility of the LBM for different calculation conditions, several test examples are examined. For the refractive index matched boundary case, the effects of the angle of the incidence on temporal signals of transmittance and reflectance are studied. 2. Mathematical formulation Consider a plane-parallel slab filled with a grey medium of finite thickness L as shown in Fig. 1(a). The medium is assumed to have azimuthal symmetry with constant physical properties. The refractive index of the medium n is homogeneous and could be either equal to or higher than those of the environment (in this paper, the refractive index of the environment is 1). The boundary at x = 0 is exposed to a collimated short-pulse irradiation with an incident angle θ0 and a radiation intensity I0 as illustrated in Fig. 1(a). The wave shape of the short pulse could be either square or Gaussian as shown in Fig. 1(b). Besides, it is assumed that the thermal emission of the medium is negligible as compared with the incident radiation. The propagation of light pulse in the semi-transparent medium is described by the transient radiative transfer equation (TRTE). In the Cartesian coordinate system, the TRTE for onedimensional problem can be written as

σ n ∂I ( x, μ , t ) ∂I ( x, μ , t ) +μ = − β I ( x, μ , t ) + s ∂t ∂x 2 co



1

−1

I ( x, μ , t )Φ ( μ , μ ′)d μ ′.

(1)

where co is the propagation speed of light in vacuum, n is the refractive index of the medium, t is the time, μ is the direction cosine of the polar angle (−1 ≤ μ ≤ 1), I(x, μ, t) is the radiative intensity, and κa, σs and β = κa + σs are the absorption, scattering and extinction coefficients, respectively. Φ(μ, μ′) = 1 + aμμ′ is the scattering phase function with a denoting the anisotropically scattering coefficient.


θ0

x L

−t p

−2t p

(a)

0

tp

2t p

3t p

(b)

Fig. 1. (a) A planar medium subjected to collimated short-pulse radiation with an incident angle θ0, and (b) the shape of the short pulse.

When n = 1, the collimated radiation penetrates directly into the medium without changing the direction. In this case the intensity I within the medium can be divided into two components, viz., the collimated intensity Ic and the diffuse intensity Id. I = Ic + Id . (2) The collimated component Ic within the medium is decreased exponentially according to Beer’s law [23] dI c = −β Ic . ds Substituting Eqs. (2) and (3) in Eq. (1) yields

(3)

∂I ( x, μ , t ) n ∂I d ( x, μ , t ) +μ d = − β I d ( x, μ , t ) + S d + S c = − β I d ( x, μ , t ) + St . ∂t ∂x co

σs



1

σs



1

I c ( x, μ , t )Φ ( μ , μ ′)d μ ′ are the source 2 2 terms resulting from the diffuse and the collimated components of radiation, respectively. St = Sc + Sd is the total source term. As the refractive index of the medium matches with the environment at the interface (with refractive index n = 1), the two boundaries at x = 0 and x = L are considered to be nonreflecting. Thus the radiative boundary condition can be written as

where Sd =

−1

I d ( x, μ , t )Φ ( μ , μ ′)d μ ′ and Sc =

(4)

−1

I wm = 0. (5) When the refractive index mismatches with the environment at the surface (n > 1) and the medium surfaces are supposed to be diffusely reflecting and semitransparent, the transient process of radiation transfer is quite different from the case of n = 1. The collimated pulse irradiation on the left boundary is divided into two parts, viz., the diffusely reflected intensity towards the environment and the diffusely transmitted intensity towards the medium inside. Therefore, only the diffuse intensity Id can be obtained. The external diffuse reflectivity ρO can be expressed as [38]: 1 2

ρ o ( n ′) = +

(3n′ + 1)( n′ − 1) n′2 ( n ′2 − 1) 2 (n ′ − 1) + ln 6( n ′ + 1) 2 ( n ′2 + 1)3 (n ′ + 1)

2n ′3 (n ′2 + 2n ′ − 1) 8n ′4 ( n ′4 + 1) ln(n ′). − + ( n ′2 + 1)( n ′4 − 1) ( n ′2 + 1)( n′4 − 1) 2

(6)


where n′ = n /1 = n. Considering the effect of total reflection, the internal diffuse reflectivity is [38] 1 [1 − ρ o ( n′)]. (7) n ′2 Thus for the diffusely reflecting semitransparent surface, the radiative boundary condition can be expressed readily as

ρ I (n ′) = 1 −

I wm = (1 − ρ I )n 2 I 0 +

ρI π



n w ⋅ s m′ > 0

n w ⋅ s m′ I wm wm′ , n w ⋅ sm < 0.

(8)

According to Eq. (7), the (1−ρI ) n2I0 could be replaced by (1−ρO) I0. In this paper, both the square and Gaussian pulses irradiation is considered. For the square pulse, the radiation intensity incident on the boundary at x = 0 may then be written as I cS (0, μ , t ) = I 0 [ H (t ) − H (t − t p )]δ ( μ − μ0 ).

(9)

where tp is the pulse duration, H(t) is the Heaviside step function, δ is the Dirac delta function, and μ0 is the direction cosine of the angle of incidence. While for the Gaussian pulse irradiation, the incident radiation intensity is expressed as

I cG (0, μ , t ) = I 0 exp[ −4 ln 2 × (

t − tc 2 ) ]δ ( μ − μ0 ). tp

(10)

The temporal Gaussian pulse shape described in Fig. 1(b) is a truncated Gaussian distribution with the maximum at t = tc and the half maxima at t = tc ± tp/2, where the pulse intensity exceeds one-half of the maximum intensity. As for n = 1, the collimated intensity Ic in the medium for the square pulse and the Gaussian pulse are needed and can be derived from Eqs. (3), (9) and (10). The collimated remnant of the square pulse irradiation can be expressed as I cS ( x, μ , t * ) = I 0 exp( − β s ) ×  H (t * − β s ) − H (t * − β s − t *p )  × δ ( μ − μ0 ).

(11)

where s = x/μ0 is the geometric distance in the incident direction, t* = βcot and tp* = βcotp is the non-dimensional time and the non-dimensional pulse-width, respectively. The attenuation of the collimated Gaussian irradiation as it travels through the medium is given by I cG ( x, μ , t * ) = I 0 exp( − β s ) exp[ −4 ln 2 × (

t * − β s − tc* 2 ) ]δ ( μ − μ0 ). t *p

(12)

In terms of the non-dimensional time t*, the RTE given by Eq. (4) is now rewritten as ∂I d ∂I + μ d + β I d = St . (13) * ∂t ∂x Using fully implicit backward differencing scheme in time, Eq. (13) can be written as nβ

nβ

I d − Id ∂I + μ d + β I d = St . Δt * ∂x

(14)

where Id is the radiative intensity value of the last time step calculation and Δt * is the nondimensional time step. Equation (14) can be expressed in a simplified form as

μ

∂I d β nβ + I d = St + * Id . ∂x B Δt

(15)


Δt * . n + Δt * The above equation can be rearranged as

where B =

∂I d 1 nβ β (16) = ( St + * Id − I d ). ∂x μ Δt B For the LB method, a pseudo time marching is performed with a Μ − velocity lattice model in 1D (D1QM) [37]. M is the total number of discrete directions. It is obviously that the speed of particle propagation along the mth discrete direction is em = ( Δx μ m ) Δt * . Using the spatial finite difference to discretize the left side of Eq. (16), the TRTE in the mth discrete direction can be expressed as

I dm ( x + em Δt * , t * + Δt * ) − I dm ( x, t * ) 1 nβ β = ( St + * Id − I d ) Δx Δt B μ nB 1β B = ( S (t * ) + * Id m − I dm ( x, t * )). μB β t Δt m = 1, 2,..., M The evolution equation corresponding to Eq. (17) is given as I dm ( x + em Δt * , t * + Δt * ) = I dm ( x, t * ) +

(17)

 Δx β  B n m St ( t * ) + I − I dm ( x, t * )   m * d μ B β n + Δt 

n m β B  = I ( x, t ) + Δt em  St (t * ) + I − I dm ( x, t * )  . * d B β n + Δt  m d

*

(18)

*

Using the standard LBM terminology [37], Eq. (17) can be rewritten as I dm ( x + em Δt * , t * + Δt * ) = I dm ( x, t * ) +

eq Δt *  m I d ( x, t * )} − I dm ( x, t * )  . {    τm

where the τm is the relaxation time for the collision process and {I dm }

eq

(19)

is the equilibrium

particle distribution function. The relaxation time τm is, therefore, calculated as

τm =

B em β

(20)

.

and, by comparing Eq. (19) with Eq. (18), the equilibrium particle distribution function can be expressed as n m Id . (21) n + Δt * β Since the polar angle space is divided equally into M parts, the source term Sc and Sd are computed from the following equations:

{I

Sc = Sd =

σs 2

σs 2



1



1

−1

−1

m d

( x, t * )} = eq

B

St ( t * ) +

I c ( x, μ , t )Φ ( μ , μ ′)d μ ′ = I d ( x, μ , t )Φ ( μ , μ ′)d μ ′ =

σs 2

σs 2

M

I

m ' =1

m' c

M

I

m ' =1

Φ m ' mω m ' .

m' d

Φ m ' mω m ' .

(22a) (22b)

where ω m ' is the angular weight of direction m′ .


For one-dimensional slab, the time-resolved signals at the boundary of incidence and opposite are termed as reflectance qR and transmittance qT, respectively. In the analysis of the transient radiative transfer, the time-resolved reflectance and transmittance provide specific information about the media. Transmittance is defined as the dimensionless net radiative heat flux emerging out of the medium due to transmission, namely the dimensionless net radiative heat flux at the right boundary (x = L). Reflectance is the dimensionless net radiative heat flux at the boundary which is subjected to the laser irradiation, and in the present case, it is the dimensionless reflected heat flux at the left boundary (x = 0). For n = 1, the time-resolved signals can be expressed as qR ( t * ) =

2π  I d (0, μ , t * ) μ d μ q0

μ < 0.

,

(2π  I ( L, μ, t )μd μ + I ( L, t ) cosθ ) , q (t ) = *

*

d

*

T

c

0

q0

(23a)

μ > 0.

(23b)

where q0 = I 0 cos θ 0 . For n >1, they can be defined as qR ( t * ) =

πρ o I 0 + 2π (1 − ρ I )  I d (0, μ , t * ) μ d μ

qT (t * ) =

q0

2π (1 − ρ I )

(  I ( L, μ, t )μd μ ) ,

,

μ < 0.

(24a)

*

d

q0

μ > 0.

(24b)

where q0 = π I 0 . After the derivation of the D1QM lattice Boltzmann model for one-dimensional transient radiative transfer is completed, the implementation of LBM solution process can be carried out according to the following routine. Step 1: Set the initial parameters, using appropriate number of lattices to mesh the solution domain. Step 2: Confirm the time step Δt* and total calculation time span. Step 3: Loop at each time step. (1) Loop for the global iterations. (a) For each discrete direction m, implement the streaming and colliding processes according to Eq. (19), and update the radiative intensity. (b) Impose boundary conditions on the boundary nodes. (c) Terminate the global iteration process if the stop criterion (the maximum relative error of source term St is not bigger than a very small value) is satisfied. Otherwise, go back to step (a). (2) Compute the time-resolved reflectance and transmittance from Eqs. (23a) and (23b) or from Eqs. (24a) and (24b), respectively. If the total non-dimensional time reaches the total calculation time span, terminate the iteration process of the time loop, otherwise, go back to step (1).


3. Results and discussion

In the following section, to verify the accuracy of the LBM approach for solving transient radiative transfer problems in absorbing, scattering media under the irradiation of short-pulse lasers, the LBM formulation is validated firstly, and the investigation of its computational efficiency is also conducted. Following that, several test examples are shown. In all these cases, the geometric thickness of the 1-D medium is L = 1.0 m. At a given time level, convergence is assumed to have been achieved when the change in source term St value at all points for the two consecutive iterations do not exceed 1 × 10−7. The present LBM for transient radiative heat transfer is coded using MATLAB. All runs were taken on Intel(R) Core(TM) i5-2320 processor with 3.00GHz CPU and 6GB RAM. 3.1 The correctness and computational efficiency of the LBM In this case, the isotropic scattering medium is contained in a slab with the optical thickness τL = 1 and the scattering albedo is ω = 1. The pulse incident on the left side of the slab is a square pulse with duration tp* = 1.0. With normal incidence of the collimated radiation (μ0 = 0.0), the LBM results for the transient transmittance signal qT are shown in Fig. 2. The nondimensional computation time span is taken as t* = 10. For grid and ray independent solutions, a maximum of 201 lattices and 12 directions are used. The time interval is chosen as Δt* = 0.05. We can see that the LBM results agree with those obtained by FVM [23] very well.

0.6

Transmittance qT

LBM FVM Ref. [23]

0.4

-1

β=1m ω = 1.0

0.2

0.0

0

2

4

*

Time, t

6

8

10

Fig. 2. Comparison of the time-resolved signals of transmittance by LBM with those by FVM [23] for β = 1.0 m−1 and ω = 1.0

Under the same calculation parameters, the CPU times (in second) taken by the LBM and FVM for different number of rays, lattices and non-dimensional time interval are presented in Table 1. It is seen from Table1 that the computational time used by the LBM is always less than the FVM. It can be concluded that the LBM is computationally efficient.


Table 1. Computational efficiency of the LBM CPU times (s) LBM FVM 59.0876 109.0126 53.6119 99.2675 40.0033 88.6963

No. of lattices/control volumes 201 201 201

No. of rays 12 12 12

Non-dimensional time step 0.04 0.05 0.1

101 301 501

12 12 12

0.1 0.1 0.1

25.8089 52.8983 80.3616

47.4902 97.2501 141.6579

201 201 201

8 14 20

0.1 0.1 0.1

31.3055 43.1718 54.4317

79.8971 95.9052 113.0984

3.2 Transient radiative transfer in isotropically scattering medium with short square pulse laser irratiation In this case, the propagation of short square pulse laser in homogenous, isotropically scattering media is solved by the LBM. The time-resolved signals of transmittance and reflectance for one-dimensional slab are obtained, and the results are compared with those in the literature. For tp* = 1.0, with normal incidence of the collimated radiation (μ0 = 0.0), the curves of qT and qR have been plotted in Figs. 3(a)-3(f) for different scattering albedo and extinction coefficient. The effects of scattering albedo ω, taken as 0.5 and 0.9, on the results are shown for extinction coefficient β = 1, 5 and 10 m−1, respectively. It can be seen that the transmittance signals begin to appear just at t* = τL and the reflectance signals remain available since the start of the transient process. For a given extinction coefficient β, with decreases in ω, the peaks of both the signals decrease and they last for a shorter duration. In all these cases, number of 12 rays and the time interval chosen as Δt* = 0.05 are considered. These solutions have been obtained with lattices of 201 for β = 1 m−1, 501 for β = 5 m−1 and 1001 for β = 10.0 m−1. It can be seen that for β = 1 m−1 and 5 m−1 with different scattering albedo, results by the LBM agree well with those obtained by the FVM [23]. In Fig. 3(e), for β = 10.0 m−1 and ω = 0.9, an obviously deviation from the FVM and DTM solutions is found. While the LBM results agree well with those by Monte Carlo method (MCM) developed by ourselves.


0.125 ω = 0.9

0.4

LBM FVM Ref. [23]

0.5

-1

β=1m

0.2 0.5

0

2

0.075

4 6 * Time, t

8

0.050

0.000

10

ω = 0.5

0

2

4 6 * Time, t

(a)

10

0.125 ω = 0.9

LBM FVM Ref. [23]

0.012

-1

β=5m

0.008

0.9

0.004

5

10

0.075

-1

β=5m ω = 0.9

0.050 0.025

ω = 0.5

0

LBM FVM Ref. [23]

0.100 Reflectance qR

Transmittance qT

8

(b)

0.016

0.000

-1

β=1m

ω = 0.9

0.025

ω = 0.9

0.0

LBM FVM Ref. [23]

0.100 Reflectance qR

Transmittance qT

0.6

15 * 20 Time, t

25

30

0.000

0.5

0

5

(c)

Transmittance qT

0.0002

-1

β = 10 m

0.0001 ω = 0.5

0.0000

0

10

20 30 * Time, t

(e)

25

30

(d)

LBM MCM DTM Ref. [23] FVM Ref. [23]

ω = 0.9

15 20 * Time, t

40

50

0.12

Reflectance qR

0.0003

10

0.09

LBM FVM Ref. [23]

ω = 0.9

0.06

-1

β = 10 m

0.03 0.00

0.5

0

10

20 30 * Time, t

40

50

(f)

Fig. 3. Comparison of the time-resolved signals of transmittance and reflectance by LBM with those by FVM [23] for different values of the scattering albedo ω and the extinction coefficient β.

Effect of the angle of incidence θ0 on qT and qR has been shown in Figs. 4(a)-4(f). For ω = 1 and tp* = 1.0, the transmittance and reflectance signals are presented for three values of β, 1.0, 5.0 and 10.0 m−1, respectively. For each value of β, results have been illustrated for θ0 = 0°, 45° and 60°. Mishra et al. have investigated these problems [23], while the correctness of their results has not been verified. Here, for validation of the model with an oblique incidence of the pulse laser built by LBM, we compare the results for ω = 1, β = 1.0 m−1 and θ0 = 60° obtained by LBM with those by Monte Carlo method (MCM) developed by ourselves. In Figs. 4(a) and 4(b), it can been seen that the LBM results agree well with those by MCM.


0.6

0.20

Reflectance qR

45°

0.10 0.05

45 °

0.4

-1

60°

β = 1.0 m ω = 1.0

β = 1 .0 m ω = 1.0

60 °

0.2

-1

θ0 = 0°

0

2

4 * Time, t (a)

6

0.0

8

θ0 = 0°

0.1 Reflectance qR

θ0 = 45° θ0 = 60° -1

β = 5.0 m ω = 1.0

0.01 0

5

10 * Time, t (c)

15

20

4 6 * Time, t (b)

0.015

θ0 = 0°

0.010

45°

0.000

0

10

θ0 = 60°

10

LBM β = 5.0 m ω = 1.0

20

-1

30 * Time, t

40

50

60

(d) θ0 = 0°

-1

β = 10.0 m ω = 1.0

45°

Transmittance qT

β = 10.0 m ω = 1.0

8

60°

0.005

0.0020

θ0 = 45°

-1

Reflectance qR

2

0.0025 θ0 = 0°

0.1

0

0.020

Transmittance qT

0.00

Transmittance qT

0.15

MCM LBM LBM

θ0 = 0°

MCM LBM LBM

0.0015

60°

0.0010

0.01

0.0005

1E-3

0

5

10

15

20 * 25 Time, t (e)

30

35

40

0.0000

0

20

40 60 * Time, t (f)

80

100

Fig. 4. Effects of the angle of incidence on the time-resolved reflectance and transmittance.

The differences of the incident angle will change the path of the collimated light propagation, which makes influence to the time-resolved signals. It can be seen in Figs. 4(a), 4(c) and 4(e), with a bigger incident angle, the curve of the time-resolved reflectance in the time range of t* = 0 to t = tp* increases faster to a maximum value at t* = tp*, as a result of which, a higher peak value is obtained. The contribution of the pulse irradiation to the transient process is embodied in the source terms in Eq. (4). It can be concluded from the analytical solutions of the collimated intensity (see Eq. (11)) that as the incident angle increases, the source terms at all the position decrease, and as a result of which, the diffuse intensity Id obtained decreases. Thus, the numerator in Eq. (22a) decreases. However, the denominator ( q0 = I 0 cos θ 0 ) also decreases at the same time. It can be concluded from Figs. 4(a), 4(c) and 4(e) that as the incident angle increases, the denominator decreases more than the numerator. It can be further observed that, after a period of time, the curves for different


incident angle intersect with each other. The reason for this is that, for a bigger incident angle, the energy produced by pulse irradiation kept in the slab is lower, as a result of which after the time t = tp* the time-resolved reflectance decreases more quickly. It can be seen from Figs. 4(b), 4(d) and 4(f) that, for all the incident angle, the qT signals begin to appear at t* = τL, while qR signals remain available from the start of the process. For the cases of oblique incidence, the existence of the qT signals during the time period t* = τL to ts* = τL / cosθ0 is owing to the contribution of the diffuse radiation which reaches the right boundary before the collimated radiation. It can be further observed that the qT signals undergo a noticeable change at ts* and t* = ts* + tp*. This behavior is owing to the fact that the collimated radiation combined with the diffuse radiation passes through the right boundary. Right after t* = ts* + tp*, only the diffuse radiation is at work. 3.3 Transient radiative transfer in anisotropically scattering medium with square pulse laser irratiation In this case, the medium is contained in a slab with the optical thickness τL = 10 and the albedo ω = 0.998. The incident pulse with normal incidence on the left side of the slab is a square signal with a duration of tp* = 1.0. Three values of the anisotropically scattering coefficient a are taken for the case. Backward scattering is considered for a = – 0.9, isotropic scattering for a = 0.0 and forward scattering for a = + 0.9. Lu et al. [10] investigated this problem by Reverse Monte Carlo Method (RMCM). The LBM results for the time-resolved reflectance and transmittance are presented in Fig. 5. For obtaining the convergent and stable solutions, a combination of 1001 lattices and 22 equally spaced directions are required in this case. The results are plotted for time span of t* = 100. With the time interval setting as Δt* = 0.1, it takes about 389.899 CPU seconds for the calculations. It can be seen that in Fig. 5(a) the time-resolved results of transmittance obtained by LBM agree well with the data obtained by the RMCM [10]. It can been observed in Fig. 5(a), the maximum of transmittance signal is higher for the case of forward scattering, while its duration in time is longer for backward scattering. It is owing to the fact that, for the case of forward scattering, photons are pushed towards the emergence wall, while for case of backward scattering, photons travel for a longer time in the medium. 1 LBM RMCM Ref. [10]

a = + 0.9

0.004 0.003

-1

0.002 0.001 0.000

β = 10 m , ω =0.998

a = 0.0

20

40 60 * Time, t

(a)

LBM a = + 0.9 a = − 0.9 a = 0.0

0.1 0.01

1E-3

a = − 0.9 0

Reflectance qR

Transmittance qT

0.005

80

100

1E-4

0

20

40 60 * Time, t

80

100

(b)

Fig. 5. The time-resolved reflectance and transmittance for three values of the anisotropically scattering coefficient: (a) the transmittance, and (b) the reflectance. The optical thickness is τL = 10 and the albedo is ω = 0.998.

3.4 Transient radiative transfer in purely scattering medium subjected to Gaussian pulse In this case, the left boundary (at x = 0) of the slab is exposed normally to a laser Gaussian pulse with an incident radiation:


2   t − 3t p     H (t ) − H (t − 6t p )  . I (t ) = I 0 exp  −4 ln 2 ×  (25)  t     p     The optical thickness of the slab is τL = 1.0 and the albedo is ω = 1.0. The LBM is used to solve the time-resolved reflectance and transmittance for the case of tp* = 0.4. Lattices of 101 and 12 directions are used in this case. With the time step taken as Δt* = 0.1, the CPU time cost for the calculation till t* = 10 is 17.73661 seconds. The results are shown in Figs. 6(a) and 6(b). The solutions obtained by LBM agree well with the data obtained by the time shift and superposition method in combination with the DFEM [25].

0.5

0.06 LBM Liu Ref. [25]

0.3

ω = 1.0 β=1m

0.2

-1

0.1 0.0

LBM Liu Ref. [25]

0.05 Reflectance qR

Transmittance qT

0.4

0.04

ω = 1.0 -1

β=1m

0.03 0.02 0.01

0

2

4 6 * Time, t

(a)

8

10

0.00

0

2

4 6 * Time, t

8

10

(b)

Fig. 6. The time-resolved transmittance and reflectance for the Gauss pulse (a) the transmittance, and (b) the reflectance.

3.5 Transient radiative transfer in two-layer nonhomogeneous media with short square pulse irradiation As shown in Fig. 7(a), the short square pulse light with width of tp* = 0.3 irradiates normally on the left surface of the two-layer isotropic scattering media. In this case, the optical thicknesses of the two-layer media are all 0.5. By using the reverse Monte Carlo method, Lu and Hsu [11] investigated the transient radiative transfer problems with the scattering albedo of ω1 = 0.1 and ω2 = 0.9, or ω1 = 0.9 and ω2 = 0.1. In their work, the geometry length is set as L1 = L2 = 0.5 mm. In [22], Ruan et al. presented a new non-dimensional number ζ (ζ = ctp/L = tp*/τ). The authors pointed out that, as the scattering albedo and the refractive index are kept the same, the temporal signals of the medium would overlap one another having different combinations of the pulse duration and the thickness of the medium with the same ζ . It means that only if τ is chosen and the non-dimensional pulse-width tp* is the same, the temporal signals are kept unchanged with different geometry length. Therefore, the geometry length L1 = L2 = 0.5 m is used in this paper. The LBM results for time-resolved reflectance are shown in Fig. 7(b). The nondimensional time is used as horizontal ordinate. Here, 401 lattices and 14 equally spaced directions are considered. The total observed time span is t* = 6. With the time interval setting as Δt* = 0.025, it takes about 72.3591 CPU seconds for the calculations. As shown in Fig. 7(b), our results are in good agreement compared with those in [11]. It can be observed from Fig. 7(b) that in the case of ω1 = 0.1 and ω2 = 0.9, the special ‘dual peak’ phenomenon occurs in the reflectance signals. Due to the strong scattering of the second layer, the local minimum in reflectance signal can be found at t* = 1 moment. Just as is presented by Lu and Hsu [11].


0.1

τ 1 =0.5 ω1

τ2=0.5 ω2

L

L

Reflectance qR

ω1=0.9, ω2=0.1

LBM RMCM Ref. [11]

0.01

ω1=0.1, ω2=0.9

1E-3

1E-4

L

0

1

2

3 * Time, t

4

5

6

(b)

(a)

Fig. 7. (a) The model of the two-layer media irradiated by the short square pulse laser and (b) the time-resolved signal of reflectance for the two-layer media.

3.6 Transient radiative transfer in isotropically scattering media with refractive index mismatched at the boundary In this case, the transient radiative transfer problem in a slab containing isotropically scattering medium with refractive-index mismatched boundary (n >1) is investigated. The two surfaces of the slab are diffusely reflecting and semitransparent. The normal incident pulse on the left side of the slab is a square pulse with duration of tp* = 1.0. In Fig. 8, the time-resolved reflectance and transmittance are plotted for the cases with n = 1.5, β = 1.0 m−1 and ω = 1.0. The MCM results developed by ourselves are also presented for the comparison. It can be seen that the LBM results agree well with those obtained by the MCM. 0.15

0.14 MCM LBM

0.10 -1

0.06

0.06

0.04

0.03

0.02 0.00

-1

β=1m ω = 1.0 n = 1.5

0.09

β=1m ω = 1.0 n = 1.5

0.08

MCM LBM

0.12 Reflectance qR

Transmittance qT

0.12

0

2

4 6 * Time, t

(a)

8

10

0.00

0

2

4 6 * Time, t (b)

8

10

Fig. 8. The time-resolved transmittance and reflectance for the case with n = 1.5, β = 1.0 m−1 and ω = 1.0, (a) the transmittance, and (b) the reflectance.

The distributions of the diffuse intensities at different directions and different nondimensional time t* are presented in Figs. 9(a) and 9(b) for x = 0 and L, respectively. In Fig. 8(a), it can be observed that, the transmittance signal begins to appear just at t* = τLn (1.5). It is owing to the fact that, the diffuse radiation produced by the normal irradiation on the semitransparent surface takes t* = τLn at least to reach the right boundary. The qT increases gradually to the peak value at the time t* = τLn + tp* (2.5). The noticeable change observed in the case of n = 1 doesn’t appear in the case of n = 1.5. This is due to the fact that in the case with the refractive index bigger than one, only diffuse radiation transfers in the


medium, and consequently the corresponding transmittance signal changes gradually inside the slab. Referring to Fig. 9(b), we can have an intuitive feeling for the increase of the qT. It can be observed that the intensities in the positive directions are great during this period of time t* = 1.5~2.5. 0.05 0.25 0.45 0.65 0.85

1 0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

μ 0

μ 0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1 0

1

2

3

t*

(a)

4

0.01 0.10 0.19 0.28 0.37 0.46

1

5

6

-1

0

3

6

t*

9

12

15

(b)

Fig. 9. Distributions of the diffuse intensities at different directions and different nondimensional time t* at (a) x = 0, and (b) x = L.

It can be observed in Fig. 8(b), the reflectance signal qR is available from the start of the transient process. During the time span of t* = 1 to t = tp* it increases gradually, and its values are noticeable higher than those after t = tp*. This is because the reflectance signals during this period consist of two parts, viz., the diffuse radiation transmitting through the internal surface and the radiation diffusely reflected by the external surface. Owing to the existence of the pulse irradiation, qR increases during this period of time. Right after the time t = tp*, only the diffusely transmitted radiation make contributions to the qR. It can be further observed that, qR increases again during the time interval of t* ≈3 to t* ≈4.5. It is owing to the fact that, the diffuse energy produced by the pulse on the left interface takes time of τLn to travel to the right interface, and after being reflected it takes a total of at least time of 2τLn (t* ≈3) to get back to the left interface . Referring to Fig. 9(a), we can also have an intuitive feeling for the increase of the qR in this period. It can be observed that the values of intensities in the negative directions are noticeable during this period. With β = 1.0 m−1 and ω = 1.0, effects of the refractive index n on the time-resolved reflectance and transmittance have been shown in Figs. 10(a) and 10(b). With different n, the diffuse reflectivity on the semitransparent surface and the propagation speed of the light in the medium are different. Consequently, the time-resolved signals for different n are different. The internal and external diffuse reflectivity of the semitransparent surface are presented in Table 2 for n = 1.2, 1.5 and 1.8. It can be seen that, with increases in n, both the internal and external diffuse reflectivity increase. It means that, for a higher n, pulsed energy transmitted through the left internal interface is lower and the energy inside the slab is reflected more by the internal interface.


0.25

0.18 n = 1.2 n = 1.5 n = 1.8

0.12

0.15

-1

β=1m ω = 1.0

0.09 0.06

-1

β=1m ω = 1.0

0.10 0.05

0.03 0.00

n = 1.2 n = 1.5 n = 1.8

0.20

Transmittance qT

Reflectance qR

0.15

0

5

10

15 * 20 Time, t

25

30

0.00

0

5

10

15 * 20 Time, t

25

30

(b)

(a)

Fig. 10. Effects of the refractive index on the transient signals, (a) the reflectance, and (b) the transmittance. Table 2. Internal and External Diffuse Reflectivity on the Semitransparent Surface for n = 1.2, 1.5 and 1.8 Refractive index n = 1.2 n = 1.5 n = 1.8

ρI 0.3363 0.5963 0.7327

ρO 0.0443 0.0918 0.1341

From Fig. 10(a) we can observe the following trends of the time-resolved reflectance. (i) In the time range of t* = 0 to t* = tp*, the qR curve for the case with a higher n is of higher values but has a lower increasing rate. As it can be seen in Eq. (24a) that the ρO is added to the qR as a constant value during this period. For a higher n, the value of ρO is higher, which results in the higher value of the qR curve. The rise of the curve is owing to the diffuse radiation inside the slab. For a higher n, based on the above analysis, the diffuse radiative energy inside the slab is lower, which results in the lower increasing rate of the qR curve. (ii) After the time of tp*, a second peak value of qR appears for different n. For a higher n, the peak appears later, which is owing to the slower speed of light. (iii) For the higher n, the diffuse radiative energy inside the slab is lower. However, the curve descends slower than those cases having lower n after the second peak. It is owing to the fact that, the inner diffuse reflectivity is higher for a higher n, which means that the energy could be kept longer inside the slab. From Fig. 10(b) the following trends for the qT may be observed. (i) As expected, the qT begins to appear just at t* = τLn. (ii) With a lower n, the peak of the curve is higher. For a lower n, both the ρO and ρI are lower, and as a result of which, more pulsed energy is transmitted through the two interfaces of the slab. (ii) Similar to the curve of qR, with a higher n, the qT curve decreases slower than those cases having lower n after the peak. 4. Conclusion

The application of the LBM was extended to solve transient radiative transfer problem in a one-dimensional slab of participating medium subjected to collimated short pulse irradiation. Both the cases for refractive index matched (n = 1) or mismatched (n>1) semitransparent boundary were considered. The accuracy and efficiency of this algorithm are studied. To show the flexibility of the LBM for different working conditions, we investigated the effects of the incident angle, scattering properties, pulse laser shapes and optical inhomogeneity on transmittance and reflectance signals. Results of the LBM were found to compare very well with those data from the published literatures. In brief, the extended LBM proposed in this paper is a simple


and accurate solution scheme for the one-dimensional transient radiative transfer. Finally, the LBM was further extended to study the transient radiative transfer in homogeneous medium with a refractive index discontinuity irradiated by the short pulse laser. We observed and analyzed several interesting trends on the time-resolved signals different from those of the case considering refractive index matched boundary. Acknowledgments

This work was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51121004) and the National Natural Science Foundation of China (Grant No. 51176040).


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