On the time relaxed Monte Carlo computations for the flow over a flat nano-plate

Computers and Fluids 160 (2018) 219–229

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On the time relaxed Monte Carlo computations for the flow over a flat nano-plate M. Eskandari, S.S. Nourazar∗ Department of Mechanical Engineering,Amirkabir University of Technology, 424 Hafez Ave, Tehran, P.O. Box 15875-4413, Iran

a r t i c l e

i n f o

Article history: Received 21 April 2017 Revised 10 September 2017 Accepted 20 October 2017 Available online 21 October 2017 Keywords: Boltzmann equation Direct simulation Monte Carlo Time relaxed Monte Carlo Higher order collision Nano-plate,

a b s t r a c t In the present study, the first, second and third orders of the TRMC scheme (TRMC1, TRMC2 and TRMC3 schemes) are employed to numerically investigate the Argon flow over a flat nano-plate with different free stream velocities and variety of Knudsen numbers. The simulations cover flow regimes from early slip to early transition regimes (0.00129 ≤ Kn ≤ 0.09). Higher order terms in the Wild sum expansion are considered to obtain higher order collisions. The results are compared with those from the standard DSMC method. Comparisons show that among the studied schemes, the results obtained from the TRMC3 scheme have excellent agreement with the ones from the DSMC method. On the other hand, the results of the TRMC1 and TRMC2 schemes show deviations compared to those from the DSMC method. The deviations are more pronounced for the temperature and pressure distributions. Moreover, the present investigation illustrates that as the Knudsen number increases the accuracy of lower orders of the TRMC scheme improves. It is observed that truncating the Wild sum expansion up to the third order approximation of the TRMC scheme, may be a proper alternative method for the DSMC method in simulating the flow over the nano-plate for Knudsen numbers 0.00129 ≤ Kn ≤ 0.09 with reasonable accuracy and simplicity in mathematics. © 2017 Published by Elsevier Ltd.

1. Introduction Navier–Stokes equations lose the capability of simulating the flow accurately, when the characteristic length of the flow is comparable to the mean free path. For such flows, the governing equation is the Boltzmann equation of kinetic theory [1,2]. During the past decades, the direct simulation Monte Carlo (DSMC) method has been extensively adopted for the numerical solution of the Boltzmann equation in rarefied regimes [1–3]. Despite the simplicity and reasonable accuracy of the DSMC method, high computational expenses of the flow simulations is the main concern. The computational expenses of the DSMC method drastically increase as the Knudsen number decreases, where the expenses of the DSMC method cannot be justified for the flows near the continuum region. Hence, introducing schemes to reduce the computational expenses of numerical solution of the Boltzmann equation, are highly interested [3–7]. Recently, the time relaxed Monte Carlo (TRMC) method has been introduced as a simple and efficient method to numerically solve the Boltzmann equation in the

∗

Corresponding author. E-mail addresses: [email protected] (M. Eskandari), [email protected] (S.S. Nourazar). https://doi.org/10.1016/j.compfluid.2017.10.021 0045-7930/© 2017 Published by Elsevier Ltd.

flows with wide variation of Knudsen numbers. In the algorithm of the TRMC scheme, time discretizations are obtained from the Wild [8] sum expansion with the higher order collisions being replaced by the local Maxwellian distribution [9]. It is illustrated that the TRMC method performs in the same way as the standard DSMC method for large Knudsen numbers, while local Maxwellian distribution replaces the time consuming collision calculations, as the Knudsen number decreases. Furthermore, the capability of adopting larger time steps compared to the ones in the DSMC method is another advantages of the TRMC scheme over the standard DSMC method [3]. On the other hand, complexity of the TRMC scheme, especially for higher order schemes, is the main drawback of the scheme. Pareschi and Russo [10] performed the stability analysis on the TRMC scheme and proved the A-stability and L-stability of the scheme. Moreover, using the TRMC scheme they obtained reasonable solution of the Kac equation compared to the results from standard DSMC method. Furthermore, Pareschi et al. [11–13] introduced algorithm for the TRMC scheme using the variable hard sphere model for inter-molecular collisions. They also studied one-dimensional shock wave problem and obtained reasonable results compared to the ones from the standard DSMC method. Pareschi and Trazzi [3] presented algorithms to obtain the first and second orders of the TRMC scheme that simultaneously preserve the conserva-

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Nomenclature

μ ρ τ F f Kn L M P Q T t U

v

the mean collision frequency, kinematic viscosity density relaxed time, shear stress transformed probability distribution function probability distribution function Knudsen number characteristic length local Maxwellian distribution collisional operator, pressure collisional operator temperature time velocity velocity vector

Subscripts ∞ mps x, y

free stream condition most probable molecular thermal speed x and y directions, respectively

tion of mass, momentum, and energy. They also simulated the gas flow around an obstacle using the TRMC method and obtained an appreciable reduction in the CPU time, compared to that from the standard DSMC method. Foreseeing higher accuracies, Russo et al. [14] introduced the third order TRMC scheme by considering higher order collisions in truncating the Wild sum expansion. They performed the TRMC and DSMC methods to investigate the Couette flow for a wide variety of Knudsen numbers and several wall velocities. The comparisons indicated that the density, velocity and temperature profiles, obtained from the TRMC scheme, are in excellent agreement with the ones from the DSMC method. Moreover, Ganjaei and Nourazar [15] studied Argon and Helium mixture flow inside a rotating cylinder using the TRMC and DSMC methods. They used Dalton’s law for partial pressures of each species and showed that the results of TRMC method are in excellent agreement with the analytical solution. They reached more accurate results using the TRMC scheme, compared to the results from standard DSMC scheme. Trazzi et al. [16] introduced the recursive TRMC scheme to obtain uniform accuracy in time, independent of time step. They obtained considerable improvement in the computational efficiency of simulating the space homogeneous test cases and non-homogeneous stationary shock problem, in comparison to the standard DSMC method. Most recently, Dimarco and Parechi [17] decomposed the collision operator of the Boltzmann equation into an equilibrium and a non-equilibrium parts and introduced the exponential Runge–Kutta scheme. They performed the first and second orders of integrating factor (IF) scheme to study the heat flux in a space homogeneous shock wave problem. They obtained results with excellent agreement with the ones from the standard DSMC method for Kn = 0.001, while both the first and second orders of IF scheme showed significant error for the flow with Kn = 0.0 0 01. Furthermore, Eskandari and Nourazar [18] performed the first, second and third orders of the TRMC scheme to study a lid-driven micro cavity flow with different lid velocities and Knudsen numbers. They obtained results with excellent agreement with the ones from the standard DSMC method, using the third order time relaxed Monte Carlo (TRMC3) scheme. 1.1. The purpose of the present work The present work focuses on investigating the truncation effects of the Wild sum expansion on the accuracy of the

TRMC scheme. Three orders of the TRMC scheme called TRMC1, TRMC2 and TRMC3 schemes are considered accordingly. Argon flow over a flat nano-plate with different free stream velocities and several Knudsen numbers is considered as the benchmark problem. The Knudsen numbers cover flows from early slip to early transition regimes (0.00129 ≤ Kn ≤ 0.09). The obtained results are compared with the ones either from the standard DSMC method or the results reported from the hybrid DSMC-NS scheme [19]. 2. Mathematical description 2.1. The Boltzmann equation The Boltzmann equation for a single component mono-atomic dilute gas in the absence of external forces, is described by the following equation, where the bilinear operator Q(f, f) describes the intermolecular collisions [1,2]:

∂f 1 + υ · ∇x f = Q ( f, f ). ∂t Kn

(1)

Considering Q ( f, f ) = P ( f, f ) − μ f, the Boltzmann equation takes the form below [5,8,20]:

∂f 1 + υ · ∇x f = (P ( f, f ) − μ f ). ∂t Kn

(2)

P(f, f) is a symmetric bilinear operator describing the collision effects of the molecules and the parameter μ = 0 is the mean collision frequency. It is common to split the Boltzmann Eq: (2) into the equation of pure convection step (i.e. Q(f, f) ≡ 0) (3) and the equation of collision step (i.e. υ · ∇x f ≡ 0) (4) [1,3,9,16,21].

∂f + υ · ∇x f = 0. ∂t

(3)

∂f 1 = (P ( f, f ) − μ f ). ∂ t Kn

(4)

The equation of convection step (3) can be directly solved leaving only concerns about the equation of collision step (4). 2.2. The DSMC approach The DSMC approach is simply obtained by applying the Euler upwind scheme to the equation of collision step (4):



f n+1 =

1−

μt Kn





fn +

 μt P ( f, f ) . Kn μ

(5)

Mathematical equation of the DSMC method (5) can be probabilistically interpreted in the following way: In order to sample a particle from f n+1 , it shall be sampled from fn with probability of (1 − μt/Kn ) and shall be sampled from P(f, f)/μ with probability of μt/Kn. Since probability of event can accept neither negative values nor values larger than unity, the time step in the DSMC method shall be selected with the following consideration: 0 ≤ (μt/Kn) ≤ 1 [3]. 2.3. The TRMC approach The relaxed time τ and the transformed probability distribution function F (v, τ ) are described as the followings [3,9]:

τ = (1 − e−μt/Kn ).

(6)

F (v, τ ) = f (v, τ )e−μt/Kn .

(7)

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Rearranging Eq. (8), using Eq. (9) yields: ∞ ∞  ∂F  = kτ k−1 fk (v ) = (k + 1 )τ k fk+1 (v ). ∂τ k=0 k=0

P (F , F ) = P

∞ 

τ f k ( v ), k

k=0

∞ 

(10)

τ f k (v ) k

k=0

= P ( f 0 , f 0 ) + 2τ P ( f 0 , f 1 ) + τ 2 ( 2P ( f 0 , f 2 ) + P ( f 1 , f 1 ) ) + · · · (11) Fig. 1. The flat nano-plate geometry and imposed boundary conditions.

Equating corresponding powers for the relaxation time, yields the following recursive algorithm for the coefficients fk :

Then by applying variables (6) and (7), Eq. (4) can be written as:

f (v, t ) = e−μt/Kn

∞   k=0



μ

(8)

τ k f k ( v ).

k=0

P ( fk , fk−h )

.

(12)

The TRMC scheme can be obtained by applying the Maxwellian truncation to Eq. (9):

The Cauchy problem (8) has a power series solution in the following form:

F ( v, τ ) =



h=0

∂F 1 = P ( F , F ). ∂τ μ F ( v, τ = 0 ) = f ( v, 0 ).

∞ 

fk+1

k 1  = k+1

1 − e−μt/Kn

k



f k (v ) .

(9)

f n+1 (v ) = e(−



μt Kn

)

k m   μt 1 − e(− Kn ) fk (v ) k=0

+ 1 − e(

−μt Kn

)

m+1

M ( v ).

(13)

Eq. (13) together with Eq. (5) indicate that for large Knudsen numbers (i.e. free molecular regimes) the TRMC scheme behaves in the same way as the standard DSMC scheme. As the Knudsen number decreases, the local Maxwellian distribution gradually replaces the time consuming collision terms. Furthermore, for very small

Fig. 2. Grid independency tests.

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Fig. 3. The results for Kn = 0.00129 (left), Kn = 0.009 (center) and Kn = 0.09 (right). From top to bottom, temperature distributions over the nano-plate, temperature distributions normal to the plate at x/L = 0.2 and x/L = 0.8 for case I (U∞ = 14 m/s).

Knudsen numbers (i.e. the fluid limit) the probability function approaches the local Maxwellian distribution [3].

f n+1 = f n .

lim

μt/Kn→ 0

lim

μt/Kn→ ∞

f n+1 = M (v ).

f

(v ) =

m 

(Ak fk (v )) + Am+1 M (v ).

(15)

The weight functions in the present work are selected in accordance with the ones introduced by Pareschi [3,10] as the following:

Ak = ( 1 − τ )τ k . m 

Ak − Am+1 .

k=0

Am+1 = τ m+2 .

(16)

The selected weight functions are non-negative functions which simultaneously satisfy consistency (17), conservation (18) and asymptotic preservation (19).

lim

A0 ( τ )

lim Ak (τ ) = 0,

τ →1

∀k = 0, · · · , m.

lim Am+1 (τ ) = 1.

(17)

(19)

τ →1

The main advantage of the TRMC scheme over the standard DSMC method, is replacing the time consuming collision terms with the local Maxwellian distribution. Furthermore, the TRMC scheme is capable to accept larger time steps compared to those from the DSMC method. It is noticeable that the time step is probabilistically limited in the DSMC method (see Eq. (5)), while theoretically there is no limitation for time steps in the TRMC scheme [3]. However, lower order schemes yield inaccurate results for larger time steps [18]. On the other hand, computational complexity of the higher order schemes, is the main drawback of the higher order TRMC schemes. Hence, in the present study the first three orders of the TRMC scheme (TRMC1, TRMC2 and TRMC3 schemes) are considered accordingly. Considering m = 1 in the general form of the TRMC scheme (15) with the weight functions (16), yields the first order of the TRMC (TRMC1) scheme as the followings:

 P ( f0 , f0 ) + τ 3 M (v ) μ = A0 f n + A1 f 1 + A2 M ( v ).

f n+1 = (1 − τ ) f0 +

= 1.

τ Ak ( τ ) lim = 0, ∀k = 1 , · · · , m + 1 . τ →0 τ τ →0

(18)

(14)

k=0

Am = 1 −

Ak ( τ ) = 1.

k=0

It is noticeable that different weight functions may also be used in the described scheme. A general form of the TRMC scheme can be obtained by using different weight functions, while the coefficients fk are defined by Eq. (12) [3,9–13,16]: n+1

m +1 



τ − τ3

(20)

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Fig. 4. The results for Kn = 0.00129 (left), Kn = 0.009 (center) and Kn=0.09 (right). From top to bottom, velocity and temperature distributions over the nano-plate, temperature distributions normal to the plate at x/L = 0.2 and x/L = 0.8 for case II (U∞ = 141 m/s).

m = 2 results in the second order of the TRMC (TRMC2) scheme:

 P ( f0 , f0 ) μ  2  P ( f , f ) 0 1 + τ − τ4 + τ 4 M (v ) μ = A0 f n + A1 f 1 + A2 f 2 + A3 M ( v ).

f n+1 = (1 − τ ) f0 +



τ − τ2

(21)

Consequently m = 3 yields the third order TRMC (TRMC3) scheme:

 P ( f0 , f0 )  2  P ( f0 , f1 ) + τ − τ3 τ − τ2 μ μ    3  2P ( f 0 , f 2 ) + P ( f 1 , f 1 ) + τ − τ5 + τ 5 M (v ) 3μ

f n+1 = (1 − τ ) f0 +



= A0 f n + A1 f 1 + A2 f 2 + A3 f 3 + A4 M ( v ).

ii. Collides with a particle extracted from fn (a particle with no previous collision) with probability of A1 . iii. Collides with a particle extracted from f1 (a particle involved in one collision before) with probability of A2 . iv. Faces two collisions with probability of 1/3A3 as the following: for the first time collides with a particle extracted from fn and for the second time collides with a particle extracted from f1 . v. Collides with a particle extracted from f2 (a particle involved in two collisions before) with probability of 2/3A3 . vi. Is replaced by a particle sampled from a local Maxwellian distribution function with probability of A4 . Eq. (22) can also be expressed throughout the following algorithm:

(22)

Eq. (22) can be probabilistically interpreted in the following way [14]: A particle chosen at nth time step: i. Does not collide with other particles with probability of A0 .

3. The geometry and calculation condition In order to investigate the Maxwellian truncation effects of the Wild sum expansion on the accuracy of the TRMC scheme, i.e. the accuracy of TRMC1, TRMC2 and TRMC3 schemes, Argon flow over a flat nano-plate is considered in the present study.

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Algorithm 1 (TRMC3). 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 35: 36: 37: 38: 39: 40: 41: 42:

Third order time relaxed Monte Carlo scheme

compute the initial velocity of the particles for t = 0 to ntot t do mark all particles with the “0-collision” label estimate an upper bound for the cross-section σ set τ = (1 − e−μt/Kn ) compute A1 (τ ), A2 (τ ), A3 (τ ), A4 (τ ) set N1 = [NA1 /2], N2 = [NA2 /4], N3 = [NA3 /3], N4 = [NA4 ] select (N1 + N2 + N3 ) dummy collision pairs among all particles for (N1 + N2 + N3 ) pairs do compute σi j = σ (|vi − v j | ) generate random number 0 ≤  ≤ 1 if  < (σi j /σ ) then perform the collision between particles i and j compute the post collision velocities update the labels of participant particles to “1-collision” end if end for select 2N2 particles among “0-collision” and 2N2 particles among “1-collision” particles for 2N2 pairs do perform the collision between pairs compute the post collision velocities update the labels of participant particles to “1-collision” and “2-collision” end for select N3 /3 pairs among “1-collision” particles for N3 /3 pairs do perform the collision between pairs compute the post collision velocities update the labels of participant particles to “2-collision” end for select N3 /3 particles among “0-collision” and N3 /3 particles among “2-collision” particles for N3 /3 pairs do perform the collision between pairs compute the post collision velocities update labels of participant particles to “1-collision” and “3-collision” end for for N4 particles do replace particles with samples from the Maxwellian, with the same total energy and moment end for for (N − 2N1 − 4N2 − 3N3 − N4 ) particles do the energy and momentum will not change for remained particles end for end for

Fig. 1 presents the configuration of the nano-plate geometry and the imposed boundary conditions. Free stream is considered for the inlet, outlet and upper boundaries. Surface of the nanoplate is assumed to be diffuse reflector [19,21–23], while a length equal to 10% of the plate length at the upstream of the leading edge, is considered as the specular reflector. The specular boundary condition is considered to provide more reasonable inlet velocity distribution [1,19,23–26]. The height of computational domain (h) is considered to be 60% of the plate length for all cases. The free stream flow is mono-atomic Argon gas at temperature of 300 K flowing parallel to the nano-plate in the positive x direction while the surface temperature of the nano-plate is 500 K.

Three different free stream velocities are considered in the present study for the Argon flow. The stream velocities are 14 m/s, 141 m/s and 1412 m/s for cases I, II and III, respectively. The velocities are selected to investigate the nano-plate subjected to the flows with different free stream velocities varying from low to supersonic speeds. The inlet pressure and number density of the Argon gas are 41.4 MPa and 1.0 × 1028 molecules per cubic meter, respectively. Moreover, in order to study the rarefaction effects on the flow properties, three different Knudsen numbers based on the nano-plate length are considered. The plate lengths are set to 100nm, 14.38nm and 1.43nm to yield the Knudsen numbers of 0.0 0129, 0.0 09 and 0.09, respectively. The Knudsen numbers cover reasonable range of flow regimes from early slip (i.e. Kn = 0.00129) to early transition (i.e. Kn = 0.09) regimes [1–3]. Moreover, variable hard sphere (VHS) model is considered for the molecular collisions. The reference molecular diameter, viscosity index and molecular mass are set to 4.17 × 10−17 m, 0.81 and 66.3 × 10−27 kg, respectively [1]. The time step for the DSMC method is computed considering the limitation 0 ≤ μt/Kn ≤ 1.0 (see Eq. (5)) while the time step for the TRMC method, is chosen to be a fraction of the free flow time step, Ct, where C ≤ 1 is a constant. The parameter C is set to 0.4, 0.35 and 0.18 for cases I, II and III, respectively. 3.1. The grid independency test The finer cells yield more accurate results but on the other hand increase computational expenses. Hence, a grid independency study is carried out using five grid resolutions of 300 × 60, 300 × 180, 300 × 360, 300 × 540 and 300 × 720 cells. Since variation of the flow properties in the longitudinal direction is small, the grids are more refined in vertical direction [19,27,28]. The cells are uniform in x direction while they have geometric progression in y direction. To ensure reaching results independent from grids, the velocity, pressure and temperature distributions of the Argon flow over the nano-plate, obtained from different grid resolutions are investigated. Fig. 2 indicates that the pressure distributions have less sensitivity to the cell size compared to the velocity and temperature distributions. It is observed that the distributions are not affected by the grid resolutions finer than 300 × 540. However the grid resolution of 300 × 720 is used in the present study to obtain results, independent of grid resolution. Moreover, since it is intended to put at least 20 particles in each cell, 4,40 0,0 0 0 particles are considered. 4. Investigating the results of TRMC1, TRMC2 and TRMC3 schemes In this section the results obtained from three orders of the TRMC scheme, called TRMC1, TRMC2 and TRMC3 schemes, are presented. For validation purposes, a detailed comparison between the results of the TRMC schemes and those from the standard DSMC method, carried out in the present study, and the results from the hybrid DSMC-NS scheme [19] are made. It is noticeable that in order to investigate the problem more precisely, the results normal to the nano-plate at two longitudinal distances (x/L = 0.2 and x/L = 0.8) are studied in addition to the results over the nanoplate. Fig. 3 compares the temperature distributions obtained from different orders of the TRMC scheme with those from the DSMC method for different Knudsen numbers for case I (U∞ = 14 m/s). The comparisons show excellent agreement between the results from the third order TRMC (TRMC3) scheme and the ones from the DSMC method. On the other hand, the results from the TRMC1 and TRMC2 schemes show deviations in comparison to their counterparts from the DSMC method. Moreover, it is observed that the ac-

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Fig. 5. The results for Kn = 0.00129 (left), Kn = 0.009 (center) and Kn = 0.09 (right). From top to bottom, velocity, temperature and pressure distributions over the nano-plate for case III (U∞ = 1412 m/s).

curacy of lower orders of the TRMC scheme increases as the Knudsen number increases. Fig. 3 indicates that the temperature jump on the nano-plate increases as the Knudsen number increases. This may be attributed to the rarefaction effects on the heat transfer mechanism. Furthermore, temperature distributions normal to the nano-plate indicate that as the Knudsen number decreases, the flow temperature reaches the free stream temperature, more rapidly. It may be interpreted that larger Knudsen number leads to an increase in the thermal boundary layer. Fig. 4 present the velocity and temperature distributions over the nano-plate and the temperature distributions normal to the nano-plate at different longitudinal distances for case II (U∞ = 141 m/s). The comparisons illustrate excellent agreement between the results of the TRMC3 scheme and their counterparts from the DSMC method. On the other hand, the results from the TRMC1 and TRMC2 schemes show deviations. The deviations are more pronounced for the TRMC1 scheme. Moreover, Fig. 4 shows that the accuracy of TRMC2 scheme is enhanced as the Knudsen number increases. This may be attributed to the fewer collisions in the flow with higher Knudsen numbers. Fig. 4 indicates that the velocity slip and temperature jump increases as the Knudsen number increases. This may be attributed to the rarefaction effects on the momentum and heat transfer mechanisms. Furthermore, temperature distributions normal to the plate illustrate that thermal boundary layer over the plate increases as the Knudsen number increases. Figs. 5 and 6 compare the results obtained from different orders of the TRMC scheme with their counterparts either from the DSMC

method or the ones from the hybrid DSMC-NS scheme [19]. Moreover, in order to measure the performance of the TRMC scheme more accurately, the results are also compared with the ones obtained from the first order integrating factor (IF) scheme [17], carried out in the present work. The velocity, temperature and pressure distributions over the nano-plate are presented in Fig. 5, while in order to investigate the problem more precisely, Fig. 6 focuses on the pressure and temperature distributions normal to the nanoplate at different longitudinal distances. Figs. 5 and 6 show excellent agreement between the results from the TRMC3 scheme with the ones from the DSMC, hybrid DSMC-NS and IF methods [19]. On the other hand, it is illustrated that the results from the TRMC1 and TRMC2 schemes show deviation. The deviations are more pronounced for the results from the TRMC1 scheme. Moreover, Fig. 6 illustrates that deviation in the results for lower order schemes are more pronounced near the leading edge. This may be attributed to higher gradients in the leading edge. It is observed that the accuracy of lower orders of TRMC scheme (i.e. TRMC1 and TRMC2 schemes) increases as the Knudsen number increases. This may be attributed to the fewer collisions in rarefied flows. Fig. 5 illustrates that the velocity slip and temperature jump over the nano-plate increase significantly as the Knudsen number increases. Furthermore, it is observed that the temperature and pressure distributions far from the leading edge significantly increase as the Knudsen number increases. This may be attributed to the rarefaction effects on the momentum and energy transfer mechanisms. It is noticeable that the maximum temperatures in the temperature distributions of Fig. 6 are higher than the temperature of either the nano-plate or the free stream. This

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Fig. 6. The results for Kn = 0.00129 (left), Kn = 0.009 (center) and Kn = 0.09 (right). From top to bottom, temperature and pressure distributions normal to the plate at x/L = 0.2 and x/L = 0.8 for case III (U∞ = 1412 m/s).

may be attributed to the formation of the oblique shock wave in case III which leads to high amount of energy dissipation. Since the maximum temperature at each longitudinal section occurs close to the plate, it can be interpreted that the shock wave is created very close to the surface of the nano-plate. Temperature distributions in Fig. 6 show that the flow field temperature is more affected by the nano-plate as the Knudsen number decreases. It can be interpreted that the thermal boundary layer over the nano-plate grows as the flow becomes more diluted. Figs. 3–6 illustrate that the results of TRMC1 and TRMC2 schemes show deviations when compared with the results of the standard DSMC method for selected time steps discussed in Section 3. The results show that deviations are more pronounced for the TRMC1 scheme. It is also observed that the accuracy of the results obtained from the TRMC2 scheme, compared to the ones from the standard DSMC method improves as the Knudsen number increases. This may be attributed to the lower collision rates in flows with higher Knudsen numbers. Fig. 7 presents the intermolecular collision rates for the test cases studied in the present

Fig. 7. Comparison of collision rates at different Knudsen numbers.

work. Fig. 7 illustrates that the collision rate significantly increases as the Knudsen number decreases, while the collision rate slightly increases as the free stream velocity increases.

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Table 1 Comparison of the normalized computational CPU times of the DSMC method and orders of the TRMC scheme for simulating the free stream over the nano-plate at different Knudsen numbers and different free stream velocities. Kn number

Kn = 0.00129

U∞ (m/s)

14

141

1412

Kn = 0.009 14

141

1412

Kn = 0.09 14

141

1412

DSMC TRMC1 TRMC2 TRMC3

1.0 0 0 0.872 0.930 0.940

1.0 0 0 0.695 0.745 0.774

1.0 0 0 0.491 0.526 0.547

1.0 0 0 0.878 0.921 0.956

1.0 0 0 0.799 0.843 0.859

1.0 0 0 0.686 0.706 0.725

1.0 0 0 0.985 1.013 1.024

1.0 0 0 0.970 1.003 1.020

1.0 0 0 0.943 0.985 1.015

Fig. 8. The velocity streamlines superimposed on the temperature contours for Kn = 0.00129 (left), Kn = 0.009 (center) and Kn = 0.09 (right). From top to bottom U∞ = 14, 141 and 1412m/s ,respectively.

Table 1 presents the normalized computational CPU times for different orders of the TRMC scheme and the standard DSMC method, carried out in the present work for all studied cases. It is noticeable that the CPU time of the standard DSMC method is used to normalize the CPU times of each case study. Table 1 indicates that the normalized computational expenses of the TRMC schemes for the flows near continuum region (i.e. Kn = 0.00129) are significantly lower than that for the standard DSMC method. This may be attributed to replacing the time consuming collision terms with the local Maxwellian distribution in the TRMC scheme. Furthermore, the improvement in the computational expenses of the TRMC schemes are enhanced as the stream velocity increases, which is related to wider variation of Knudsen number in the flows with higher stream velocities. Moreover, Table 1 illustrates that the improvement in the computational CPU time of the TRMC scheme decreases as the flow rarefaction increases. It is observed that the CPU time of the TRMC schemes reach to their counterparts from the standard DSMC method for Kn = 0.09, where this may be attributed to the fewer inter-molecular collisions in the rarefied flows. Table 1 indicates that up to 45% reduction in the computational CPU time, compared to the CPU time from the DSMC

method, can be obtained using by using TRMC3 scheme for simulating free stream over the nano-plate with reasonable accuracy and complexity in mathematics.

5. Comparison of the results for different free stream velocities Fig. 8 presents the velocity streamlines superimposed on the temperature contours for different free stream velocities and different Knudsen numbers, carried out in the present work from the TRMC3 scheme. It is observed that the maximum temperature of the flow increases significantly as the free stream velocity increases. The increase in the temperature for case I (U∞ = 14 m/s) is negligible while the temperature rise for case III (U∞ = 1412 m/s) is substantial and a hot spot is created near the leading edge of the nanoplate. Creation of the hot spot may be attributed to the formation of the shock wave in case III which results in high amount of energy dissipation. Moreover, Fig. 8 demonstrates that as the Knudsen number increases the flow temperature is more affected by the hot spot. This affection is more pronounced for the flow with

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Fig. 9. The results for Kn = 0.00129 (left), Kn = 0.009 (center) and Kn = 0.09 (right). From top to bottom, shear stress and temperature distributions over the nano-plate and the contours for local Mach number for case III (U∞ = 1412 m/s).

Kn = 0.09, where the majority of the flow is affected by the hot spot. It is found that for the flows with low free stream velocities, the streamlines downstream the leading edge are more affected by the nano-plate. This may be attributed to the larger velocity boundary layer for the flows with low free stream velocities. Furthermore, the effect of the nano-plate on the streamlines increases as the Knudsen number increases which can be interpreted in the following way: For the flows with low Knudsen numbers, the particles affected by the nano-plate cannot scatter in the upstream flow because of consecutive collisions with unaffected particles of the free stream, while intermolecular collisions decreases drastically as the Knudsen number increases (see Fig. 7). Hence, the particle effect on the flow becomefs prominent as the Knudsen increases. Fig. 9 presents the dimensionless shear stress and temperature distributions over the nano-plate subjected to the free Argon stream with different Knudsen numbers and different free stream velocities. The reference shear stress is defined as τ0 = 2 , where U 0.5 × ρ∞ × Umps mps refers to the most probable molecular thermal speed [1]. The distributions show significant increase in the shear stress at the leading edge of the nano-plate for case

III (U∞ = 1412 m/s). Such increase in the shear stress, results in the high amount of energy dissipation at the leading edge. Furthermore, following the trend for the shear stress distributions, the temperature distributions illustrate abrupt increase at the leading edge of the nano-plate for case III. It is observed that the asymptotic increase in shear stress and temperature distributions along the nano-plate, increases significantly as the Knudsen number increases. It is noticeable that the abrupt increase in the temperature and shear stress distributions in case III may be attributed to the formation of oblique shock wave in case III which results in high amount of energy dissipation at the nano-plate attacking edge. Fig. 9 also indicate that the shock wave nears the plate as the Knudsen number increases. 6. Conclusion In the present work, three orders of Maxwellian truncation on the Wild sum expansion, resulting in the TRMC1, TRMC2 and TRMC3 schemes are investigated. The schemes are employed to simulate the free Argon stream over the flat nano-plate with different free stream velocities and different Knudsen numbers. The free stream velocities and Knudsen numbers are considered

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to cover the flows from low speed to supersonic stream velocities (14 m/s ≤ U∞ ≤ 1412 m/s) and the flow regimes from early slip to early transition regimes (0.00129 ≤ Kn ≤ 0.09). It is observed that the velocity, temperature and pressure distributions obtained from the TRMC3 scheme are in excellent agreement with those from either the standard DSMC method or the hybrid DSMC-NS scheme. On the other hand, the results from the TRMC1 and TRMC2 schemes show deviations compared to those from the standard DSMC method. The deviations are more pronounced for the temperature distributions. Furthermore, It is observed that as the Knudsen number increases, the accuracy of lower orders of the TRMC scheme improves. It is concluded that the TRMC3 scheme may be a suitable alternative scheme for the standard DSMC method in simulating the flow over the nano-plate for Knudsen numbers 0.00129 ≤ Kn ≤ 0.09 and arbitrary free stream velocity with reasonable accuracy and simplicity in mathematics. References [1] Bird GA. Molecular gas dynamics and the direct simulation of gas flows. Oxford,: Clarendon Press; 1994. ISBN 978-0198561958. [2] Cercignani C. The Boltzmann equation and its applications. Vol. 67 of Applied mathematical sciences. New York, NY: Springer New York; 1988. ISBN 978-14612-6995-3. doi:10.1007/978- 1- 4612- 1039- 9. arXiv: 1011.1669v3. [3] Pareschi L, Trazzi S. Numerical solution of the Boltzmann equation by time relaxed Monte Carlo (TRMC) methods. Int J Numer Methods Fluids 2005;48(9):947–83. doi:10.1002/fld.969. [4] Filbet F, Russo G. High order numerical methods for the space nonhomogeneous Boltzmann equation. J Comput Phys 2003;186(2):457–80. doi:10. 1016/S0 021-9991(03)0 0 065-2. [5] Gabetta E, Pareschi L, Toscani G. Relaxation schemes for nonlinear kinetic equations. SIAM J Numer Anal 1997;34(6):2168–94. doi:10.1137/ S0036142995287768. [6] Oran ES, Oh CK, Cybyk BZ. DIRECT SIMULATION MONTE CARLO : Recent Advances and. Ann Rev Fluid Mech 1998;30:403–41. [7] Pan LS, Liu GR, Khoo BC, Song B. A modified direct simulation Monte Carlo method for low-speed microflows. J Micromech Microeng 20 0 0;10(1):21–7. doi:10.1088/0960-1317/10/1/304. [8] Wild E. On Boltzmann’s equation in the kinetic theory of gases. Math Proc Cambridge Philos Soc 1951;47(03):602. doi:10.1017/S0305004100026992. [9] Pareschi L, Caflisch RE. An implicit Monte Carlo method for rarefied gas dynamics. J Comput Phys 1999;154(1):90–116. doi:10.1006/jcph.1999.6301. http: //linkinghub.elsevier.com/retrieve/pii/S0021999199963015. [10] Pareschi L, Russo G. Asymptotic preserving Monte Carlo methods for the Boltzmann equation. Transp Theory Stat Phys 20 0 0;29(3-5):415–30. doi:10.1080/ 0 0411450 0 08205882. [11] Pareschi L, Russo G. An introduction to Monte Carlo method for the Boltzmann equation. ESAIM: Proc 2001;10:35–75. doi:10.1051/proc:20 010 04.

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