Sruti Chigullapalli

Entropy considerations in numerical simulations of non-equilibrium rarefied flows

Abstract Non-equilibrium rarefied flows are encountered frequently in supersonic flight at high altitudes, vacuum technology and in microscale devices. Prediction of the onset of non-equilibrium is important for accurate numerical simulation of such flows. We formulate and apply the discrete version of Boltzmann’s H -theorem for analysis of non-equilibrium onset and accuracy of numerical modeling of rarefied gas flows. The numerical modeling approach is based on the deterministic solution of kinetic model equations. The numerical solution approach comprises the discrete velocity method in the velocity space and the finite volume method in the physical space with different numerical flux schemes: the first-order, the second-order minmod flux limiter and a third-order WENO schemes. The use of entropy considerations in rarefied flow simulations is illustrated for the normal shock, the Riemann and the two-dimensional shock tube problems. The entropy generation rate based on kinetic theory is shown to be a powerful indicator of the onset of non-equilibrium, accuracy of numerical solution as well as the compatibility of boundary conditions for both steady and unsteady problems.

Uncertainty in microscale gas damping: Implications on dynamics of capacitive MEMS switches

Effects of uncertainties in gas damping models, geometry and mechanical properties on the dynamics of micro-electro-mechanical systems (MEMS) capacitive switch are studied. A sample of typical capacitive switches has been fabricated and characterized at Purdue University. High-fidelity simulations of gas damping on planar microbeams are developed and verified under relevant conditions. This and other gas damping models are then applied to study the dynamics of a single closing event for switches with experimentally measured properties. It has been demonstrated that although all damping models considered predict similar damping quality factor and agree well for predictions of closing time, the models differ by a factor of two and more in predicting the impact velocity and acceleration at contact. Implications of parameter uncertainties on the key reliability-related parameters such as the pull-in voltage, closing time and impact velocity are discussed. A notable effect of uncertainty is that the nominal switch, i.e. the switch with the average properties, does not actuate at the mean actuation voltage. Additionally, the device-to-device variability leads to significant differences in dynamics. For example, the mean impact velocity for switches actuated under the 90%-actuation voltage (about 150 V), i.e. the voltage required to actuate 90% of the sample, is about 129 cm/s and increases to 173 cm/s for the 99%-actuation voltage (of about 173 V). Response surfaces of impact velocity and closing time to five input variables were constructed using the Smolyak sparse grid algorithm. The sensitivity analysis showed that impact velocity is most sensitive to the damping coefficient whereas the closing time is most affected by the geometric parameters such as gap and beam thickness.

Unsteady 3D Rarefied Flow Solver Based on Boltzmann-ESBGK Model Kinetic Equations

Formulation and verification of unsteady rarefied flow solver based on BoltzmannESBGK equations in arbitrary three-dimensional geometries is presented. The solver is based on the finite volume method in physical space and the discrete ordinate method in velocity space with an implicit time discretization. Verification is carried out for an unsteady approach to equilibrium, a steady one-dimensional Couette flow and a two-dimensional quasi-steady gas damping problem. Finally, the application of the full 3D parallel solver is considered to simulate unsteady microscale gas damping in a micro-electro-mechanical system switch.

Modeling of Viscous Shock Tube Using ES-BGK Model Kinetic Equations

The viscous effects on unsteady shock wave propagation are investigated by numerical solution of the Boltzmann model kinetic equations. The kinetic equations are solved for two unsteady non-equilibrium flow problems, namely, the one-dimensional Riemann problem and a two-dimensional viscous shock-tube. The numerical method comprises the discrete velocity method in the velocity space and the finite volume discretization in physical space using various flux schemes. The discrete version of H-theorem is applied for analysis of accuracy of the numerical solution as well as of the onset of non-equilibrium. Simulations show that the maximum entropy generation rate in viscous shock tube occurs in the boundary layer / shock wave interaction region. The entropy generation rate is used to determine the time-variation of the speed of propagation of shock, contact discontinuity and rarefaction waves.

Nonlinear effects in squeeze-film gas damping on microbeams

We consider squeeze-film gas damping during microbeam motion away and toward a substrate as it occurs during opening and closing of RF switches and other MEMS devices with moving components. The numerical solution of the gas-damping problem in two-dimensional geometries is obtained based on the Boltzmann?ES-BGK equation. The difference in damping force between a downward- and upward-moving beam with a gap-to-width ratio of 34 is shown to vary from as little as from 5% for low beam velocities of 0.1 m s?1 to more than 200% at 2.4 m s?1. For a constant velocity magnitude of 0.8 m s?1, this difference increases from 60% to almost 90% when the pressure is reduced by an order of magnitude. The numerical simulations are consistent with earlier observations of a significantly higher damping force during the closing of a capacitive RF MEMS switch reported by Steeneken et al (2005 J. Micromech. Microeng. 15 176?84). The physical mechanism leading to this nonlinear dependence of the damping force on velocity has been attributed to the differences in the flow rarefaction regime for the gas in the microgap.

Immersed Boundary Method for Boltzmann Model Kinetic Equations

Three different immersed boundary method formulations are presented for Boltzmann model kinetic equations such as Bhatnagar-Gross-Krook (BGK) and Ellipsoidal statistical Bhatnagar-Gross-Krook (ESBGK) model equations. 1D unsteady IBM solution for a moving piston is compared with the DSMC results and 2D quasi-steady microscale gas damping solutions are verified by a conformal finite volume method solver. Transient analysis for a sinusoidally moving beam is also carried out for the different pressure conditions (1 atm, 0.1 atm and 0.01 atm) corresponding to Kn=0.05,0.5 and 5. Interrelaxation method (Method 2) is shown to provide a faster convergence as compared to the traditional interpolation scheme used in continuum IBM formulations. Unsteady damping in rarefied regime is characterized by a significant phase-lag which is not captured by quasi-steady approximations.

Non-Equilibrium Flow Modeling Using High-Order Schemes for the Boltzmann Model Equations

We consider application of higher-order schemes to the Boltzmann model equations with a goal to develop a deterministic computational approach that is accurate and efficient for simulating flows involving a wide range of Knudsen numbers. The kinetic equations are solved for two non-equilibrium flow problems, namely, the structure of a normal shock wave and an unsteady two-dimensional shock tube. The numerical method comprises the discrete velocity method in the velocity space and the finite volume discretization in physical space with different numerical flux schemes: the first-order, the second-order minmod flux limiter as well as a third-order WENO scheme. The normal shock wave solutions using BGK and ES collision models are compared to the DSMC simulations. The solution for unsteady shock tube is compared to the Navier-Stokes simulations at low Knudsen numbers and the rarefaction effects in such flow are also discussed. It is observed that a higher-order flux scheme provides a better convergence rate and, hence, reduces the computational effort. The entropy generation rate is shown to be a very sensitive indicator of the onset of non-equilibrium as well as accuracy of a numerical scheme and consistency of boundary conditions in both flow problems.

Implications of Rarefied Gas Damping for RF MEMS Reliability

Capacitive and ohmic RF MEMS switches are based on micron‐sized structures moving under electrostatic force in a gaseous environment. Recent experimental measurements [4, 5] point to a critical role of gas‐phase effects on the lifetime of RF MEMS switches. In this paper, we analyze rarefied flow effects on the gas‐damping behavior of typical capacitive switches. Several damping models based on Reynolds equation [7, 8] and on Boltzmann kinetic equation [9, 6] are applied to quantify the effects of uncertainties in fabrication and operating conditions on the impact velocity of switch contact surfaces for various switch configurations. Implications of rarefied flow effects in the gas damping for design and analysis of RF MEMS devices are discussed. It has been demonstrated that although all damping models considered predict a similar damping quality factor and agree well for predictions of closing time, the models differ by a factor of two and more in predicting the impact velocity and acceleration at contac...

Knudsen Force Modeling in Application to Microsystems

At the microscale, even moderate temperature differences can result in significant Knudsen forces generated by the energy exchange between gas molecules and solids immersed in a gas. Creating, controlling and measuring Knudsen forces in microsystems can be an arduous task since only limited theory exists at present. This present study investigates the mechanism of Knudsen forces in detail based on numerical solution of the Boltzmann kinetic equation. The Knudsen force is shown, in general, to be a result of thermal nonequilibrium between gas and solid. The simulations are validated by comparison with experimental measurements that have been reported by Passian et al. 10 using heated atomic force microscope probes. A closed-form model for the Knudsen force on a beam is obtained based on the simulations and can be applied for analysis and design of microsystems.

Modeling of Microstructures Actuation by the Knudsen Thermal Force

Heated microscale objects immersed in a gas ambient are subject to thermal Knudsen forces generated by the non-equilibrium energy exchange between gas molecules and solid surfaces. Knudsen forces are significant when the length scale of a temperature gradient is comparable to the gas molecular mean free path. This can occur for very low gas pressures or at extremely small length scales. The overall goal of this work is to study the feasibility of using Knudsen force as an alternative actuation mechanism for N/MEMS. The kinetic solution of Boltzmann equation using the discrete ordinate/finite volume discretization in the high-dimensional phase space is circumventing difficulties associated with traditional stochastic DSMC approach in dealing with the slow bulk motion. The comparison to measurements by (Passian et al, PRL, 2003) shows that Knudsen force is well reproduced by simulations assuming full momentum accommodation for nitrogen and argon gas and an incomplete accommodation for helium. It provides a pathway for design and analysis of devices taking advantage of the benign mechanism of the Knudsen forces, in particular, the absence of high electric fields. The analysis shows that the Knudsen force results in an impact velocity of only 0.9 cm/s whereas electrostatic forces with low voltages below 0.5V result in impact velocity in the range of 6–20 cm/s. We further discuss how Knudsen force can be used for low impact velocity actuation and also to overcome the stiction problem.Copyright