a dsmc algorithm for gaseous flows through rough

General information GASMEMS is an International Training Network for young researchers in the field of rarefied gas flows in Micro Electro Mechanical Systems (MEMS). This 4-year project is funded by the European Community under the Seventh Framework Programme. Gas flows in microsystems are of great importance for various applications that touch almost every industrial field (e.g. fluidic microactuators for active control of aerodynamic flows, vacuum generators for extracting biological samples, mass flow and temperature micro-sensors, pressure gauges, micro heat-exchangers for the cooling of electronic components or for chemical applications). The main characteristic of gas microflows is their rarefaction, which for device design often requires modeling and simulation both by continuous and molecular approaches. The role played by the interaction between the gas and the solid device surfaces becomes essential and it is generally not well understood. Numerous models of boundary conditions have been proposed but they often require an empirical adjustment strongly dependent on the micro manufacturing technique. The GASMEMS network has been built from several existing collaborations within bilateral programmes, from scientific collaborations and national networks with the objectives:

(i) to structure research in Europe in the field of micro gas flows to improve global fundamental knowledge and enable technological applications to an industrial and commercial level; (ii) to train PhD and Post-doctoral researchers at a pan-European level, with the aim of providing both a global overview on problems linked to gas flows and heat transfer in microsystems, and advanced skills in specific domains of this research field. To achieve these objectives, yearly Workshops and Summer Schools will be organized. This CD contains the proceedings of the 1st international GASMEMS workshop in Eindhoven, the Netherlands. Within the Workshop a variety of talks are selected to present a state-of-the art view to the covered topics as well as new developments and research results. The program includes keynote lectures, invited lectures and contributed papers. We would like to thank the Scientific Committee for their work with the paper evaluation and selection process. Besides, we thank all the reviewers of the full papers for their efforts spent to make this workshop a success! Prof. Stéphane Colin, network coordinator Dr. Lucien Baldas, network coordinator assistant Dr. Arjan Frijns, editor and event coordinator

Proceedings of the 1st GASMEMS Workshop- Eindhoven, September 7-8, 2009

GASMEMS09-07

A DSMC ALGORITHM FOR GASEOUS FLOWS THROUGH ROUGH MICROCHANNELS S. K. Stefanov

Institute of Mechanics, Bulgarian Academy of Sciences, Sofia, Bulgaria

Abstract A Monte Carlo procedure for simulation of molecular flow through highly porous layer is presented. The procedure is implemented in the standard DSMC method. Its application to microfluidic problems is illustrated with results obtained from the simulation of a pressure driven gaseous flow through a plane micro-channel with rough walls.

1 Introduction In microfluidic systems the surface to volume ratio is large causing boundary effects to be considerably more important than in macroscopic devices. The roughness of the walls might increase this ratio significantly. Comparison of numerical and experimental data shows that friction characteristics of smooth and rough micro-channels are significantly different. A number of theoretical models of the wall roughness such as randomly distributed conical surfaces, periodically corrugated channel surface etc. is proposed in papers [1-7]. In the present paper the wall roughness is suggested to be simulated as a highly porous medium with randomly distributed body skeleton [8]. The present paper deals with some problems of the mathematical formulation and the computer simulation of gas flows through a fine highly porous medium presenting the wall roughness of the channel walls. An important feature of these flows is that the mean pore size of the porous medium is comparable to the mean free path of the molecules moving inside the pores. A well-known fact is that the use of continuum models for mathematical description of this class of flows leads to wrong predictions of the basic macroscopic properties of the gas fluxes through the porous layer. In this case, for a correct analysis the gas motion within the pores must be interpreted as a rarefied gas flow, and consequently, described mathematically on the base of the Boltzmann kinetic equation, see Cercignani [9]. On the other side the motion of a single gas molecule through the pore skeleton can be described by using the wellknown “dusty gas” model, which considers the porous body skeleton as a system of large, compared with the gas molecules immovable particles randomly distributed within the porous domain. Thus, the velocity distribution function of the molecules moving through the porous body skeleton can be obtained from a linear integral equation describing the scattering of the molecules from the surface of the large immovable particles. In our consideration, both processes of intermolecular collisions and molecular scattering from pore boundaries are incorporated into one kinetic equation. The obtained equation is solved by using the widely used, in rarefied gas dynamics, Direct Simulation Monte Carlo (DSMC) method, see Bird [10], appropriately modified for the aims of our analysis. Finally, an example of simulation of a pressure driven gaseous flow through a plane micro-channel with rough walls with equal temperatures is considered.

2 The basic kinetic equations A stochastic model of a porous layer can be constructed in many different manners. For simplicity, we consider the motion of a monatomic gas consisting of small rigid spheres, which are moving through a system of large immovable spheres (dusty gas) distributed randomly in the porous domain. According to 1

Proceedings of the 1st GASMEMS Workshop- Eindhoven, September 7-8, 2009

the assumptions of the kinetic theory, the gas molecules interact via binary collisions or reflect from the surface of the immovable spheres. Both collision processes can be incorporated into one kinetic equation for the velocity distribution function f (ξ, x, t ) , which is as follows: ∂f ∂f ∂f ∂f + ξ1 + ξ2 + ξ3 = Q( f , f ) + P( f ) ∂t ∂x ∂y ∂z

(1)

where x = ( x, y, z ) and ξ = (ξ1 , ξ 2 , ξ 3 ) are the position and velocity vectors of a gas molecule, while the first term of the right-hand Q( f , f ) =

∫

1 ( f ′f*′ − ff* ) B (θ, ξ − ξ * )dξ *dθdε m

(2)

is the nonlinear Boltzmann collision operator .The second term P( f ) =

1 m

[∫ K (ξ′ → ξ, x) f σ′( ξ′ , x) ξ′.n dξ′dn − ξ.n σ( ξ , x) f ]

(3)

is the linear integral operator of molecule scattering from the porous body skeleton presented by the system of immovable spheres. In (2) and (3), B(θ, ξ − ξ * ) is a kernel describing the details of binary molecular interactions and K (ξ ' → ξ, x) is the kernel describing the scattering of molecules by the porous medium, m is the gas molecular mass, f ′, f*′, f* . are the same as f (ξ, x, t ) , except ξ is replaced by ξ′, ξ′* , ξ * . The molecular velocity ξ * is an integration variable (the velocity of any molecule colliding with a molecule of velocity ξ ), while ξ′ and ξ′* are the velocities of two molecules entering a collision, which brings them to velocities ξ and ξ * for the Boltzmann collision operator, and respectively ξ′ → ξ for equation (3). The angles θ and ε give the direction along which these molecules approach each other; n is an unit vector normal to the surface of a large immovable sphere at a collision point x . Finally, we assume hard-sphere molecules, in which case B(θ, ξ − ξ* ) = d 2 ξ − ξ* cos θ sin θ , where d is the gas molecule diameter; and σ( ξ , x) is a cross-section of a collision “gas molecule-immovable particle”. The crosssection σ is related to the mean free path of gas molecules with respect to the system of the immovable spheres λ p , see Pavlyukevich et al. [11]. Further, we use the following expression for the probability of a molecule to pass a distance x to the next collision with an immovable sphere: (4) w = exp{− x / λ} . The mean free path with respect to the immovable spheres is given by λ=

2 β D, 3 1− β

(5)

where D is the diameter of the immovable spheres. The porosity β is defined as follows: β = 1−

nD (4 / 3)π( D / 2) 3 V

(6)

where nD is the number of the immovable spheres in the porous body volume V. We complete our formulation with the following general initial and boundary conditions for the velocity distribution function (see Cercignani [9]): at t = 0 f (ξ, x,0) = f 0 , x ∈ V ; (7) for t > 0 : ξ.n f (ξ, x, t ) = ∫ ' R(ξ′ → ξ, x, t ) f (ξ′, x, t ) ξ′.n dξ′ + ξ.n f e (ξ, x, t ), x ∈ ∂V . (8) ξ .n < 0

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Proceedings of the 1st GASMEMS Workshop- Eindhoven, September 7-8, 2009

where R is a kernel describing the scattering rules at the porous body boundary ∂V , for example, in the case of specularly reflecting boundary it is as follows R(ξ′ → ξ, x, t ) = δ(ξ − ξ′ + 2n n.ξ′ ) .

The source term fe is a distribution function of the entering the porous layer gas molecules. It is worth noting that changing the form of the scattering kernel K (ξ′ → ξ, x) in eq. (3) one can describe not only the scattering by porous medium consisted of immovable spheres but also scattering by cones with given aperture [2,3], or by regularly corrugated surface etc. Finally, any macroscopic quantity of the gas, such as density, bulk velocity, temperature, pressure etc., can be obtained as a linear functional (momentum) of the distribution function f (ξ, x, t ) .

3 DSMC procedure The direct simulation Monte Carlo (DSMC) technique, originally proposed by Bird [9], uses model particles that move and collide in physical space to perform a direct simulation of the molecular gas dynamics. Intermolecular collisions are modelled using stochastic rules and this distinguishes the technique from the molecular dynamics method. As showed and proved by Wagner [12], the DSMC approach can be considered as a numerical method for solving the Boltzmann equation. The DSMC algorithm given below is devised in agreement with the remarks above and appropriately modified in accordance with the formulations of section 2 for the gas motion through the porous medium. The basic steps of the simulation in a three-dimensional space domain V (the reduction to one- or two-dimensional case is trivial) are as follows: the time interval [0, T], over which the solution is sought, is subdivided into subintervals with step ∆t : (a) the space domain was subdivided into cells with sides ∆x = (∆x, ∆y, ∆z ) , (b) the gas molecules were simulated in the gap G with a stochastic system of N points having positions x i (t ) = ( xi (t ), yi (t ), zi (t )) and velocities ξ i (t ) , (c) within each time step there are Nm molecules in the mth cell; this number is varied by computing its evolution in the following two stages: Stage 1. The binary collisions in each cell are calculated without moving the particles Stage 2. All particles in the computational domain are moved at a distance proportional to their new velocities (no binary collisions in this stage). If the distance, at which each molecule is shifted, is larger than the distance to the domain boundary or larger than the computed free path (4) to the next immovable large sphere of the porous body, then an interaction with the boundary or the sphere occurs. The new molecule velocity is computed from the probability distribution derived from the corresponding kernels R and K (see equations (8) and (3), respectively), (d) stages 1 and 2 are repeated until t = T, (e) the important moments of the distribution functions are calculated by time averaging over a given large time interval. Let us now describe the two stages of the calculation mentioned in (d) in some detail. In Stage 1 we use Bird's [10] “no time counter” scheme, which envisages the following two steps: Step 1. Computation of the maximum number of binary collisions by using the formula: N c max =

N m ( N m − 1) < πd 2 > ξ i − ξ j 2βVcell

max

∆t ,

(9)

where Vcell = ∆x∆y∆z is the cell volume. In (9) the cell volume is reduced by a factor equal to the porosity β. 3

Proceedings of the 1st GASMEMS Workshop- Eindhoven, September 7-8, 2009

Step 2. Ncmax pairs (ij) of particles are randomly chosen. Each of these pairs is 'collided' with probability ξ i − ξ j / ξ i − ξ j . If the collisional event occurs, the velocities after collision are max

calculated in the following way: 1 (ξ i + ξ j + k ξ i − ξ j ) 2 1 ξ′j = (ξ i + ξ j − k ξ i − ξ j ) 2

ξ′i =

(10) (11)

where k is a vector randomly distributed on the unit sphere. Otherwise the velocities are left unchanged. Stage 2 envisages the following steps: Step 1. The new position of the molecules are computed x′i = x i + ξ i ∆t .

(12)

Step 2. The free path x to a next immovable large sphere is computed from equation (4). Step 3. If x′i − x i is larger than the distance to the domain boundary or larger than the computed free path x then an interaction with the boundary or the sphere occurs. The new molecule velocity is computed through a probability distribution derived from the corresponding kernels R and K, see equations. (8) and (3), respectively.

4 Gaseous flow through a rough microchannel We consider a pressure driven monatomic gas flow in a two-dimensional micro-channel with height H and length L (L>> H). The computational domain and the imposed local rectangular coordinate system (x, y) are shown in Fig. 1. Initially, the channel is filled with gas of density, ρ0=mn0 (m is the mass of the gas molecules and n0 is the number density) and temperature, Tw. The two channel walls and the immovable spheres have temperature Tw.The inlet and outlet pressures are equal to pin and pout , respectively. Diffuse reflection is assumed on both channel walls and on the surface of the immovable spheres presenting the porous medium.

Y

Tw H/2 Pout

Pin

L

X

Figure 1: Geometry of gas flow through a channel with rough walls presented by a porous layer with depth = δ H / 2 − Ypor For illustration of the proposed approach some numerical experiments have been carried out for a fixed Kn=0.028, pressure pin = 1.0 and pout =0.0, where pressure is scaled by ρ0 RTw . The computational domain [ L × H / 2] is covered by 800 × 80 uniform grid-cells. The initial total 4

Proceedings of the 1st GASMEMS Workshop- Eindhoven, September 7-8, 2009

0.14

β=0.8 β=0.9 β=0.95 β=0.99 β=0.999 β=1.0

0.12

number flux

0.1

0.08

0.06

0.04

0.02

0

0

0.1

0.2

0.3

0.4

0.5 1-δ

0.6

0.7

0.8

0.9

1

Figure 2: Number flux dependence on roughness depth δ for different values of porosity β .

1.4

δ=1.0 δ=0.5 δ=0.0

1.2

1.0

0.8 p/pin 0.6

0.4

0.2

0

0

0.5

1

1.5

2

2.5 x/H

3

3.5

4

4.5

5

Figure 3: Pressure profiles along centre line of the channel for different values of δ .

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Proceedings of the 1st GASMEMS Workshop- Eindhoven, September 7-8, 2009

number of simulated molecules is 106 . The aspect ratio of the channel is equal to = A L= / H 5 . In the following figures 2,3, and 4 the number flux through the channel is normalized by n0 2 RTw , distance by H .

1.5

p/p

in

1 0.5 0 0 20

50 15

100 10

y/(m.f.p./2)

150

5 0

x/(m.f.p./2)

200

Figure 4: Pressure field in the upper half of the channel ( fig 1).

The non-dimensional number flux through the channel is presented in Fig. 2 for different values of porosity β . The value 1 − δ gives y - coordinate of the roughness boundary. Thus, δ = 1 correspond to a case of filled out with porous medium channel and δ = 0 - to a channel with perfectly smooth walls. The pressure profiles along the centre line for δ = 0.0, 0.5,1.0 are shown in Fig. 3. The character of the pressure drop is changed significantly from non-linear for δ = 0.0 to linear for δ = 1.0 . The pressure field shown in Fig. 4 for δ = 0.5 illustrates the pressure jump formed at the edge between porous layer and clearance.

5 Conclusions The paper deals with some problems on the mathematical description and computational simulation of gas flows through highly porous media. The presented numerical results show that the Direct Simulation Monte Carlo method can be used successfully for solving a variety of problems of heat and mass transfer through microchannels with rough walls interpreted as porous layers with different thickness. Some validation results and comparisons to available experimental and numerical data will be presented in a forthcoming paper.

6 Acknowledgement The author would like to acknowledge the financial support provided by the EU FP7 Grant GASMEMS PITN-GA-2008-215504.

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Proceedings of the 1st GASMEMS Workshop- Eindhoven, September 7-8, 2009

References 1

Veijola, T. (2003), Model for Flow Resistance of a Rare Gas Accounting for Surface Roughness, Modeling and Simulation of Microsystems, NANOTECH 2003, San Fransisco, Vol. 2, pp. 492-495. 2 Gimelshein N., Duncan J., Lilly T., Gimelshein S., Ketsdever A. and Wysong I.( 2007), Surface Roughness Effects in Low Reynolds Number Channel Flows,. 25th International Symposium on Rarefed Gas Dynamics, Saint Petrsburg, Eds M. Ivanov and A. Rebrov, ISBN 978-5-7692.0924-6, , pp. 695-702. 3 Sawada, T., Horie, B. Y., Sugiyama, W. (1996),.Diffuse scattering of gas molecules from conical surface roughness,. Vacuum, 47, No. 6-8, pp. 795-797. 4 Sugiyama, W., Sawada, T., Yabuki, M., Chiba, Y. (2001),.Effects of surface roughness on gas flow conductance in channels estimated by conical roughness model,. Applied Surface Science 169-170, 787-791. 5 Aksenova, O.A., Khalidov, I.A. ,( 2005), Fractal And Statistical Models Of Rough Surface Interacting With Rare_ed Gas Flow,. 24th International Symposium on Rare_ed Gas Dynamics, AIP Conference Proceedings, Mellville NY, 762, pp. 993-998. 6 Aksenova, O. A. . (2004), The comparison of fractal and statistical models of surface roughness in the problem of scattering rarefed gas atoms,. Vestnik St. Petersburg University: Mathematics, Issue 1, pp. 61-66. 7 Sugiyama,W., Sawadaa, T., and Nakamori, K. (1996).Rarefied gas flow between two flat plates with two dimensional surface roughness,. Vacuum, , 47, No. 6-8, pp. 791-794. 8 S. Stefanov (2003), Direct Monte Carlo simulation of gas flows through highly porous media, In: “Current issues on heat and mass transfer in porous media”, Proc. NATO ASI, edt. D. Ingham, Published by Ovidius University press, pp. 520-528. 9 Cercignani, C. (1988) The Boltzmann Equation and its Applications, Springer Verlag, New York. 10 Bird, G. A. (1994) Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford. 11 Pavlyukevich, N.V., Gorelik, G.E., Levdansky, V.V., Leitsina, V.G., Rudin, G.I. (1995) Physical Kinetics and Transfer Processes in Phase Transitions, Begell House Inc., New York, Wallingford, U.K. 12 Wagner, W. (1992) A Convergence Proof for Bird’s Direct Simulation Monte Carlo method for the Boltzamann Equation, J. Statist. Phys. 66, 1011-1044

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