# Study of subsonic–supersonic gas flow through micro/nanoscale nozzles using unstructured DSMC solver

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s10404-010-0671-7

- Cite this article as:
- Darbandi, M. & Roohi, E. Microfluid Nanofluid (2011) 10: 321. doi:10.1007/s10404-010-0671-7

- 27 Citations
- 665 Views

## Abstract

We use an extended direct simulation Monte Carlo (DSMC) method, applicable to unstructured meshes, to numerically simulate a wide range of rarefaction regimes from subsonic to supersonic flows through micro/nanoscale converging–diverging nozzles. Our unstructured DSMC method considers a uniform distribution of particles, employs proper subcell geometry, and follows an appropriate particle tracking algorithm. Using the unstructured DSMC, we study the effects of back pressure, gas/surface interactions (diffuse/specular reflections), and Knudsen number on the flow field in micro/nanoscale nozzles. If we apply the back pressure at the nozzle outlet, a boundary layer separation occurs before the outlet and a region with reverse flow appears inside the boundary layer. Meanwhile, the core region of inviscid flow experiences multiple shock-expansion waves. In order to accurately simulate the outflow, we extend a buffer zone at the nozzle outlet. We show that a high viscous force creation in the wall boundary layer prevents any supersonic flow formation in the divergent part of the nozzle if the Knudsen number exceeds a moderate magnitude. We also show that the wall boundary layer prevents forming any normal shock in the divergent part. In reality, Mach cores would appear at the nozzle center followed by bow shocks and expansion region. We compare the current DSMC results with the solution of the Navier–Stokes equations subject to the velocity slip and temperature jump boundary conditions. We use OpenFOAM as a compressible flow solver to treat the Navier–Stokes equations.

### Keywords

Micro/nanoscale nozzlesRarefied flowSubsonic regimeSupersonic regimeDSMCUnstructured meshNavier–StokesSlip boundary conditionOpenFOAM## 1 Introduction

Attention to Micro/Nano-Electro-Mechanical Systems (MEMS-NEMS) has grown enormously in leading technologies such as micro/nano-technologies since a few years ago. This has led to the development of an increasing number of extremely small devices. As the hydrodynamic diameter of a conduit decreases to scales comparable with the mean free path of the flow particles moving through it, the continuum flow assumptions behind extracting the Navier–Stokes (NS) equations deteriorate rapidly. In other words, the gas can no longer be in thermodynamic equilibrium and a variety of rarefaction effects can take place. The weight of rarefaction can be measured by defining the flow Knudsen number, which is the ratio of gas mean free path (λ) to the conduit height, *Kn* = λ/*H* in a 2D channel flow. Different non-equilibrium regimes including slip (0.001 < *Kn* < 0.1), transition (0.1 < *Kn* < 10), and free molecular (*Kn* > 10) ones have already been categorized in micro/nanoscale geometries.

One of the basic geometries utilized in MEMS and NEMS is the converging–diverging nozzle. Performing a varying cross section, the flow in micro/nanoscale nozzle may experience different rarefaction regimes simultaneously. For example, it may experience both continuum and slip flow regimes at the convergent part while the divergent part may experience transition and free molecular regimes. Evidently, the NS equations; subject to the velocity slip and temperature jump boundary conditions, may be proposed to solve the micro/nanoscale nozzle flows. However, the encountered obstacles promote the researchers to choose the kinetic-based approaches, such as the direct simulation Monte Carlo (DSMC) (Bird 1994), to simulate flow in such small scales with a wide range of rarefaction regimes. Literature shows that DSMC has been largely used to predict the flow field inside micro/nanoscale devices such as micro/nanoscale channels (Xue et al. 2003; Wang and Li 2004; Roohi et al. 2009; Roohi and Darbandi 2009; Yang et al. 2009) and nozzles (Alexeenko et al. 2002, 2006; Louisos and Hitt 2005; Liu et al. 2006; Xie 2007). We mainly focus on the activities performed in the latter ones.

Alexeenko et al*.* (2002) simulated flow through axisymmetric and 3D micronozzles using both DSMC and NS solvers. They observed that the viscous effects would dominate the gas expansion and reduce the thrust mainly due to significant wall shear stress appearances. They investigated the effect of tangential momentum accommodation coefficient on the flow behavior and showed that the flow would depend on this coefficient weakly if it increases from 0.8 to 1. Louisos and Hitt (2005) solved the NS equations and studied the effects of micronozzle geometry on its performance. They reported a remarkable reduction in the nozzle thrust as the divergent angle exceeded over 60°. They reported that the subsonic boundary layer would restrict the flow and reduce the effective exit area. In another attempt, Alexeenko et al*.* (2006) used a coupled thermal-fluid analysis (finite-element DSMC) to study the performance of high temperature gas flow through MEMS-based nozzles. They calculated the temporal flow field variation and the nozzle temperature. In addition, they reported the operational time limit for thermally insulated and convectively cooled nozzles. Liu et al*.* (2006) simulated flow through small-scale nozzles using the DSMC and the NS equations with slip and jump boundary conditions. They studied the effects of inlet pressures, Reynolds number, and micronozzle geometry and reported that the continuum-based solutions would show obvious deviations from the DSMC results as *Kn* number exceeds 0.045. Xie (2007) simulated low Knudsen number micronozzle flows using DSMC and solving the NS equations. He measured the dependency of mass flow rate on the pressure differences. He also reported the occurrence of multiple expansion–compression waves in the divergent section.

Lin and Gadepalli (2009) used the continuum equations with no-slip and slip boundary conditions and simulated gas flows through micronozzles. They suggested a correlation between nozzle specific impulse, throat diameter, and the flow Reynolds number. They also examined the effect of different gases on the value of nozzle specific impulse. Titove and Levin (2007) proposed a collision-limiter method, i.e., equilibrium direct simulation Monte Carlo (eDSMC), to extend the DSMC simulations to high pressure small-scale nozzle and channel flows. Their simulation captured the compression waves in the nozzle, which were in good agreement with high-order Euler solutions. Xu and Zhao (2007) chose the NS equations subject to slip wall boundary conditions and simulated small-scale nozzle flow subject to different back pressures. They studied the shock structures at low Knudsen number flows. They found that the viscous effect would be the key parameter in shock wave formation within the micronozzle. Louisos et al*.* (2008) reviewed the key findings obtained from computational studies of supersonic micronozzle flow using both the continuum and kinetic-based techniques. They reported that the combination of viscous, thermal, and rarefaction effects on the micro-scale flow structure would considerably affect the supersonic flow behavior in micronozzles. They described different aspects of rarefaction effects in nozzle performance. They also reported that the thermal non-equilibrium, i.e., the delay in the rotational and vibrational energy relaxations, would reduce the performance in low Reynolds number micronozzle flows. San et al*.* (2009) studied the size and expansion ratio effects on the micronozzle flow field behavior solving the 2D augmented Burnett and the NS equations. They reported small differences in the solutions of low Knudsen number flows subject to either slip or no-slip boundary conditions. Sun et al. (2009) investigated the flow and temperature fields in free molecular micro-electro-thermal resist jet (FMMR) using a coupled DSMC and finite-volume method. They studied the effect of inlet pressure. Their results showed that the temperature of solid area would change drastically imposing different inlet pressure conditions.

The main objective of the current study is to provide a deeper understanding of the flow behavior in the micro/nanoscale converging–diverging nozzles. We use DSMC to confidently model a wide range of rarefaction flow regimes through the micro/nanoscale nozzles. We investigate the effects of back pressure, Knudsen number, and gas–surface interaction on the nozzle flow behavior. We also describe the correct position, where the back pressure boundary conditions, should be imposed at the nozzle exit. We have already validated our basic DSMC solver through simulating different micro/nanoscale geometries (Roohi et al. 2009; Roohi and Darbandi 2009). However, to analyze the nozzle flow more robustly, our basic DSMC solver has been suitably extended to unstructured grid applications. Using the compressible OpenFOAM solver (2009), we also compare the results of DSMC with the NS solutions whenever applicable.

## 2 The DSMC method

### 2.1 Basic algorithm

DSMC is a numerical approach to solve the Boltzmann equation based on direct statistical simulation of the molecular processes described by the kinetic theory (Bird 1994). It is categorized as a particle method in which each particle represents a large bulk of real gas molecules. The physics of gas is modeled through uncoupling the particles motion and their collisions. The implementation of DSMC needs breaking down the computational domain into a collection of grid cells. After fulfilling all the required molecular movements, the collisions between particles are simulated in each cell independently. In the current study, variable hard sphere (VHS) and Larsen–Borgnakke (LB) collision models are used and the collision pair is chosen based on the no-time counter method (Bird 1994). The main steps in DSMC method are to set up the initial conditions, to move and index the particles, to collide particles, and to sample the particles within cells to determine the thermodynamic properties such as temperature, density, and pressure. In addition, the communication between cells are required as one particle moves from one cell to another one. Therefore, each particle must reside in one individual cell for a sufficient long period to ensure binary collisions and to obtain the correct local molecular velocity distribution. Consequently, the DSMC time step should be smaller than the mean collision time, defined as λ/*V*_{mp}, where *V*_{mp} is the most probable speed. As a result of this restriction, the particles do not cross more than one cell during one time step and this provides suitable communication between cells.

*u*/

*a*= −d

*ρ/ρ*, where \( a = \sqrt {{\text{d}}p/{\text{d}}\rho } \) is the speed of sound and

*p*and ρ are pressure and density, respectively. Applying this definition to a boundary cell, it yields

*j*denote the quantities at the outlet boundary and the

*j*th cell adjacent to the outlet boundary, respectively. Using the characteristic wave equation, the velocity can be obtained from

### 2.2 Subcells arrangement

### 2.3 Initial particle distribution

*X*) from

*x*

_{1},

*x*

_{2}, and

*x*

_{3}are the

*x*-coordinates of cell vertices and 0 ≤ α ≤ 1 is a random number. Next, we find the intersections (

*y*

_{is1},

*y*

_{is2}) of a vertical line, which crosses two sides of one triangle. Moreover, the

*Y*position of particle is determined from

*y*

_{1},

*y*

_{2}, and

*y*

_{3}are the

*y*-coordinates of cell vertices and (

*α*+

*μ*) < 1. As shown in Fig. 2b, this formula results in more uniform distributions within the sub cells.

### 2.4 Particle tracking algorithm

**P**

_{i}

**P**

_{i+1}) and the connecting vector between the particle position and the vertices of the edge (

**P**

_{i}

**X**(

*t*+ d

*t*)) is positive. This cross product (

**β**(

**X**)) should be calculated for all the edges of cell unless resulting in a negative value. Then, the ZL algorithm must be performed for different layers of neighboring cells.

## 3 The Navier–Stokes solution

*Kn*< 0.1. In order to evaluate the accuracy of the NS equations in the solution of rarefied gas flows, we use OpenFOAM (2009) to simulate micronozzle flow. The OpenFOAM, Open Field Operation and Manipulation, is a programmable CFD toolkit licensed under the GNU General Public License. The OpenFOAM distribution contains numerous solvers and utilities and covers a wide range of problems. It is a finite-volume package designed to solve systems of differential equations in arbitrary 3D geometries. It uses a series of discrete C++ modules. We use “RhoCentralfoam” solver to simulate subsonic–supersonic nozzle flows. RhoCentralfoam is an explicit density-based solver for simulating the viscous compressible flow of perfect gases. It benefits from a Godunov-like central-upwind scheme. The space discretization has a second-order accuracy based on the reconstruction of the primitive variables of pressure, velocity, and temperature. Time integration employs the first-order (forward) Euler scheme. For more details about this solver, see Ref. O’Hare et al. (2007). In order to consider rarefaction effects on the walls, we use the first-order velocity slip and temperature jump boundary conditions on the walls as follows:

*n*indicates the normal direction, the subscripts w and g stand for wall and gas adjacent to wall, γ is the specific heat ratio,

*Pr*is the Prandtl number, σ

_{u}is the tangential momentum accommodation coefficient, and σ

_{T}is the thermal accommodation coefficient. We consider both of these coefficients unity in studying diatomic nitrogen gas flow. Meanwhile, Agrawal and Prabhu (2008) have shown that the value of tangential momentum accommodation coefficient would be 0.926 for monatomic gas studies. Alexeenko et al. (2002) showed that the dependency of nozzle flow solution to the accommodation coefficient is quite weak if this coefficient increases from 0.8 to 1.

## 4 Results and discussion

In this section, we present our DSMC results and the NS solutions for the micro/nano-nozzle flows and evaluate the accuracy of the NS solutions for simulating the rarefied internal gas flows. In this regard, we first describe our defined test cases. Second, we validate our DSMC solver for supersonic flow through micronozzles. We also perform grid independency test and compare the DSMC results with the NS solutions. Next, we study the effects of back pressure and gas–surface interaction on the flow behavior and temperature distribution in micronozzles. Eventually, we consider the physics of high Knudsen number flows through nanonozzles.

### 4.1 Test cases

*V*

_{mp}. The values of mass flow rate at the inlet and outlet are monitored until achieving negligible differences between two subsequent time steps.

*H*

_{t}is 15 μm and

*Kn*

_{in}= 7.5 × 10

^{−4}. For cases 6–7,

*H*

_{t}= 40 nm and

*Kn*

_{in}= 0.153. In all simulations,

*P*

_{in}= 1 atm and

*T*

_{wall}=

*T*

_{in}= 300 K. The Reynolds number \( \left( {\text{Re}_{\text{t}} = \rho_{\text{t}} c_{\text{s}} H_{\text{t}} /\mu } \right), \) based on the nozzle throat height and the speed of sound (

*c*

_{s}) is only reported for the viscous wall cases. The nozzle thrust is calculated using \( F_{\text{t}} = \dot{m}u, \) which is proportional to the throat Reynolds number because \( \dot{m} = \text{Re}_{\text{t}} \mu_{0} h \). Therefore, the Reynolds number at the throat, as reported in Table 1, is an indicator of the nozzle propulsion performance.

Details of the investigated micro/nano nozzle test cases

Case |
| \( \overline{Kn}_{0} \) (×10 |
| |
---|---|---|---|---|

Diffusive | Specular | |||

1 | – | 8.01 | 9.67 | 406 |

2 | 7 | 3.24 | 6.92 | 260 |

3 | 15 | 1.95 | 1.63 | 269 |

4 | 25 | 1.93 | 1.13 | 257 |

5 | 35 | 1.01 | 0.91 | 247 |

6 | – | 521 | 382 | 0.05 |

7 | 20 | 244 | – | 0.04 |

### 4.2 Validation and grid independency tests: case 1

*X*/

*L*= 0.0–0.3 considering different MPC magnitudes. These differences are due to a low speed flow near the inlet of nozzle. We previously observed that the pressure distribution would remain unchanged if we increase the MPC from 12 to 17. Le et al. (2006) also reported similar conclusions about the effects of MPC value on the accuracy of DSMC solution. We will use MPC ≥ 25 to perform our DSMC calculations.

*x*or

*y*directions. In the next stage, we present grid independent study for our extended DSMC solver using two grid resolutions of 2,747 (Grid 1) and 18,765 (Grid 2) unstructured cells. Since the region in vicinity of the wall is the most sensitive part with respect to the cell sizes, we study the velocity slip and temperature jump distributions along the nozzle wall. The results are shown in Fig. 7 and are compared with the NS solution. In the DSMC simulation, we should keep the size of the largest cells less than the local mean free path in the direction where the gradients of the flow are high, which is in the transversal direction (Le et al. 2006; Cai et al. 2000; Shen et al. 2003). The features of unstructured grid facilitate suitable grid refinement with the local mean free path variations. It should be noted that the size of cells in grid 2 is smaller than the mean free path in the transversal direction. The two DSMC solutions presented in Fig. 7 are close to each other; however, the solution for grid 2 is closer to the NS solution. Meanwhile, the scattering observed in the DSMC solutions are due to a relatively lower speed of flow near the wall. We observe slip in the velocity distribution and jump in the temperature distribution right at the beginning of the divergent part of nozzle. Agrawal et al

*.*(2005) also reported similar behavior in solving rarefied flow in channels geometry with sudden expansion or contraction. As is indicated by the reference, these discontinuities originate from the gas compressibility effects.

*.*(2006). The DSMC solutions agree with each other. It should be reminded that Liu et al

*.*(2006) used a grid with 410 × 140 non-uniform quadrilateral elements to solve this test case. A good agreement between the current DSMC solution and that of Liu et al

*.*(2006) indicates that the cell employed in the current simulation is fine enough to achieve suitable accuracy. Since the Knudsen number is small in the convergent part, both the DSMC and the NS solutions (subject to slip/jump boundary condition) agree closely there. However, the discrepancy between the NS and DSMC solutions increases as the flow expands and experiences more serious rarefaction in the divergent part. As is seen, the NS solution without applying suitable slip/jump boundary conditions is far incorrect, especially for the temperature distribution.

*Kn*= λ/

*H*

_{out}). It is observed that the gas experiences one-order of magnitude rarefaction as the flow is expanded in the divergent part. The rarefaction is slightly stronger near the nozzle walls since the velocity acceleration due to slip flow and the wall heat transfer reduces the density magnitude. In the vicinity of the nozzle exit, the impact of flow rarefaction is significant again. As indicated in Table 1, the average outlet Knudsen number is about 8.01 × 10

^{−3}, which is lower than the slip flow limit. However, the local Knudsen number at the outlet increases well above the slip flow limit, see Fig. 12a. This description can provide good reason about the differences between the NS and DSMC solutions observed in Figs. 9, 10, 11. Figure 12b shows the pressure contours. It is observed that the pressure distribution is non-uniform along the

*y*-direction, i.e., d

*p*/d

*y*> 0 in the convergent part and d

*p*/d

*y*< 0 in the divergent part. Uribe and Garcia (1999) have also reported non-uniform pressure distribution in the rarefied Poiseuille flow. Figure 12c shows the velocity vector field. We can clearly observe a rapid expansion of the rarefied flow in the divergent part.

### 4.3 Effects of back pressure

#### 4.3.1 Role of boundary layer

#### 4.3.2 Specular walls

In Table 1, there are two columns of data for the average Knudsen number at the outlet. The first column is for viscous (diffusive) wall and the second one is for the inviscid (specular) wall simulations. Since Knudsen number inversely depends on the density magnitude, a higher outlet Knudsen number ratifies a higher velocity and a lower pressure/temperature there. For example, cases 1 and 2 with specular wall boundary condition show higher Mach and lower density values at the outlet (see Fig. 16) compared with the same cases imposing diffusive wall boundary condition (see Fig. 14). Therefore, the Knudsen number for cases 1 and 2 with specular boundary condition are higher than the diffusive one. Conversely, the average outlet Mach number for specular condition is lower than that of diffusive walls for cases 3–5. Therefore, the average Knudsen number is lower for the specular walls in cases 3–5.

#### 4.3.3 Temperature profiles

*P*

_{back}= 15 kPa, which is due to a Mach core appearance around

*X*/

*L*= 0.4. The case with

*P*

_{back}= 25 kPa shows slight heating at the centerline due to flow expansion. As is seen, the temperature is constant in the separated region in this frame; however, it increases in Fig. 17c as the flow exits the outlet. The decrease in temperature is stronger for lower back pressures, which is due to having a stronger expansion there.

### 4.4 Flow in nanonozzles

*Kn*

_{in}= 0.153 to 0.244. They are obtained under different boundary conditions. Figure 18a refers to a pure supersonic flow in the nozzle considering diffusive (viscous) walls. Figure 18b refers to a pure supersonic flow in the nozzle with specular (inviscid) walls. Figure 18c depicts Mach contours for a nozzle flow with

*P*

_{back}= 20 kPa, case 7. The average outlet Knudsen number, based on the outlet height, is

*Kn*

_{o}= 0.521 in Fig. 18a,

*Kn*

_{out}= 0.382 in Fig. 18b, and

*Kn*

_{out}= 0.244 in Fig. 18c. Figure 18a shows a particular behavior, i.e., it is observed that the flow accelerates normally in the divergent section and that the Mach number reaches a value of unity at the buffer zone exit. The acceleration of subsonic flow in the divergent section is quite unexpected. In fact, it is impossible to establish supersonic flow at this high inlet Knudsen number condition. As the Knudsen number increases, the viscous forces dominate and that the dissipation of the kinetic energy becomes sufficiently high. In other words, the flow can neither be choked at the throat nor be accelerated to a supersonic condition in the divergent section. To confirm the role of viscosity, we simulate the same case, however, we impose the specular (inviscid) wall boundary conditions this time; see Fig. 18b. It is observed that the flow is choked at the throat and is accelerated in the divergent part. In addition, a series of bow shocks appears at the divergent part. These bow shocks only change higher supersonic flow conditions to weaker supersonic flow cases.

Figure 18c shows Mach contours for the case with a back pressure of 20 kPa. The flow reaches a maximum Mach number of 0.45 at the throat and is not choked. The subsonic flow decelerates in the divergent part, which is a quite physical behavior for the subsonic flow through a nozzle. Therefore, we conclude that the thick viscous boundary layers prevent the formation of supersonic flow at higher inlet Knudsen numbers in the divergent part of nanonozzles. The observed physics implies that the flow must be subsonic in the nozzle and this requires applying a back pressure at the end of nozzle. It should be mentioned that we observed similar results for flows with higher inlet Knudsen numbers. Based on the current achievements obtained from both the NS and DSMC solvers, the current authors are to extend their NS (Darbandi and Schneider 2000; Darbandi et al. 2008) and DSMC (Roohi et al. 2009; Roohi and Darbandi 2009) solvers to a hybrid NS–DSMC solver.

## 5 Conclusion

We developed an unstructured DSMC solver and simulated the subsonic and supersonic flows in micro/nanoscale converging–diverging nozzles. It was observed that the mixed impacts of rarefaction, compressibility, and viscous forces would form the flow behavior in the micro/nanoscale nozzles. The use of a buffer zone far from the nozzle exit allowed eliminating the nonphysical impact of a uniform pressure at the nozzle exit. If we apply a back pressure at the outlet, high viscous forces prevent the formation of normal shocks; alternatively, the regions of high Mach number can appear in the domain. These regions diminish due to formation of bow shocks. In this case, the flow passes through an approximately constant height conduit rather than a divergent nozzle shape because a significant portion of the flow separates near the wall. If we eliminate the flow viscosity from the walls, we observe that the thick bow shocks would appear normal to the walls. Meanwhile, the separated region exists because the main flow is still viscous. We observed that it is impossible to set up supersonic flow in nanonozzles as soon as the inlet Knudsen number exceeds a moderate value. This phenomenon is due to strong viscous force appearances. We showed that it is required to apply a back pressure at the outlet to obtain a physical solution in nanonozzles.

## Acknowledgments

The authors would like to thank the Graduate Study Office of Sharif University of Technology for financial supports. Some simulations presented in this work were performed during the visit of Ehsan Roohi at the University of Strathclyde, Glasgow, UK. The authors would like to thank Iranian Ministry of Science, Research, and Technology for providing financial supports for this visit.