Duncan A. Lockerby

The usefulness of higher-order constitutive relations for describing the Knudsen layer

The Knudsen layer is an important rarefaction phenomenon in gas flows in and around microdevices. Its accurate and efficient modeling is of critical importance in the design of such systems and in predicting their performance. In this paper we investigate the potential that higher-order continuum equations may have to model the Knudsen layer, and compare their predictions to high-accuracy DSMC (direct simulation Monte Carlo) data, as well as a standard result from kinetic theory. We find that, for a benchmark case, the most common higher-order continuum equation sets (Grads 13 moment, Burnett, and super-Burnett equations) cannot capture the Knudsen layer. Variants of these equation families have, however, been proposed and some of them can qualitatively describe the Knudsen layer structure. To make quantitative comparisons, we obtain additional boundary conditions (needed for unique solutions to the higher-order equations) from kinetic theory. However, we find the quantitative agreement with kinetic theory and DSMC data is only slight.

New directions in fluid dynamics: non-equilibrium aerodynamic and microsystem flows

Fluid flows that do not have local equilibrium are characteristic of some of the new frontiers in engineering and technology, for example, high–speed high–altitude aerodynamics and the development of micrometre–sized fluid pumps, turbines and other devices. However, this area of fluid dynamics is poorly understood from both the experimental and simulation perspectives, which hampers the progress of these technologies. This paper reviews some of the recent developments in experimental techniques and modelling methods for non–equilibrium gas flows, examining their advantages and drawbacks. We also present new results from our computational investigations into both hypersonic and microsystem flows using two distinct numerical methodologies: the direct simulation Monte Carlo method and extended hydrodynamics. While the direct simulation approach produces excellent results and is used widely, extended hydrodynamics is not as well developed but is a promising candidate for future more complex simulations. Finally, we discuss some of the other situations where these simulation methods could be usefully applied, and look to the future of numerical tools for non–equilibrium flows.

On the modelling of isothermal gas flows at the microscale

This paper makes two new propositions regarding the modelling of rarefied (nonequilibrium) isothermal gas flows at the microscale. The first is a new test case for benchmarking high-order, or extended, hydrodynamic models for these flows. This standing time-varying shear-wave problem does not require boundary conditions to be specified at a solid surface, so is useful for assessing whether fluid models can capture rarefaction effects in the bulk flow. We assess a number of different proposed extended hydrodynamic models, and we find the R13 equations perform the best in this case. Our second proposition is a simple technique for introducing non-equilibrium effects caused by the presence of solid surfaces into the computational fluid dynamics framework. By combining a new model for slip boundary conditions with a near-wall scaling of the Navier–Stokes constitutive relations, we obtain a model that is much more accurate at higher Knudsen numbers than the conventional second-order slip model. We show that this provides good results for combined Couette/Poiseuille flow, and that the model can predict the stress/strain-rate inversion that is evident from molecular simulations. The model’s generality to non-planar geometries is demonstrated by examining low-speed flow around a micro-sphere. It shows a marked improvement over conventional predictions of the drag on the sphere, although there are some questions regarding its stability at the highest Knudsen numbers.

Capturing the Knudsen Layer in Continuum-Fluid Models of Nonequilibrium Gas Flows

In hypersonic aerodynamics and microflow device design, the momentum and energy fluxes to solid surfaces are often of critical importance. However, these depend on the characteristics of the Knudsen layer - the region of local non-equilibrium existing up to one or two molecular mean free paths from the wall in any gas flow near a surface. While the Knudsen layer has been investigated extensively using kinetic theory, the ability to capture it within a continuum-fluid formulation (in conjunction with slip boundary conditions) suitable for current computational fluid dynamics toolboxes would offer distinct and practical computational advantages.

Modeling and Design of Microjet Actuators

A computational model is developed to aid the design of microelectromechanical systems (MEMS) for use in active turbulence control. The focus here is on micro-actuators and, in particular, a design employed by syntheticjet devices. This consists of a diaphragm within a cavity that, by its piezoinduced motion, creates an ejection of fluid through an orifice in the cavity’s lid. The diaphragm is modeled using classical thin-plate theory, with the stiffness of the attached piezodevice incorporated. For numerical economy, the fluid motion within the cavity is not modeled; instead, the pressure is calculated with the perfect gas law. However, in the orifice, where viscous forces are more dominant, one-dimensional Navier‐Stokes equations are solved. The actuator system is modeled in its entirety. All that is required to calculate the outlet jet velocity is the input voltage applied to the piezodevice. The numerical model is validated against experimental data for synthetic-jet devices and used to predict their optimal dimensions. An alternative mode of forcing the diaphragm is proposed that does not suffer from the drawbacks inherent in synthetic-jet operation at MEMS scale. This mode generates a jump in cavity pressure, creating a pufflike jet disturbance. This concept is explored with the aim of uncovering practical issues and simple design guidelines.

Water transport through carbon nanotubes with defects

Non-equilibrium molecular dynamics simulations are performed to investigate how changing the number of structural defects in the wall of a (7,7) single-walled carbon nanotube (CNT) affects water transport and internal fluid dynamics. Structural defects are modelled as vacancy sites (missing carbon atoms). We find that, while fluid flow rates exceed continuum expectations, increasing numbers of defects lead to significant reductions in fluid velocity and mass flow rate. The inclusion of such defects causes a reduction in the water density inside the nanotubes and disrupts the nearly frictionless water transport commonly attributed to CNTs.

Hybrid continuum-molecular modelling of multiscale internal gas flows

We develop and apply an efficient multiscale method for simulating a large class of low-speed internal rarefied gas flows. The method is an extension of the hybrid atomistic-continuum approach proposed by Borg et al. (2013) 28] for the simulation of micro/nano flows of high-aspect ratio. The major new extensions are: (1) incorporation of fluid compressibility; (2) implementation using the direct simulation Monte Carlo (DSMC) method for dilute rarefied gas flows, and (3) application to a broader range of geometries, including periodic, non-periodic, pressure-driven, gravity-driven and shear-driven internal flows. The multiscale method is applied to micro-scale gas flows through a periodic converging-diverging channel (driven by an external acceleration) and a non-periodic channel with a bend (driven by a pressure difference), as well as the flow between two eccentric cylinders (with the inner rotating relative to the outer). In all these cases there exists a wide variation of Knudsen number within the geometries, as well as substantial compressibility despite the Mach number being very low. For validation purposes, our multiscale simulation results are compared to those obtained from full-scale DSMC simulations: very close agreement is obtained in all cases for all flow variables considered. Our multiscale simulation is an order of magnitude more computationally efficient than the full-scale DSMC for the first and second test cases, and two orders of magnitude more efficient for the third case.

Fluid simulations with atomistic resolution

We present a new hybrid method for simulating dense fluid systems that exhibit multiscale behaviour, in particular, systems in which a Navier-Stokes model may not be valid in parts of the computational domain. We apply molecular dynamics as a local microscopic refinement for correcting the Navier-Stokes constitutive approximation in the bulk of the domain, as well as providing a direct measurement of velocity slip at bounding surfaces. Our hybrid approach differs from existing techniques, such as the heterogeneous multiscale method (HMM), in some fundamental respects. In our method, the individual molecular solvers, which provide information to the macro model, are not coupled with the continuum grid at nodes (i.e. point-wise coupling), instead coupling occurs over distributed heterogeneous fields (here referred to as field-wise coupling). This affords two major advantages. Whereas point-wise coupled HMM is limited to regions of flow that are highly scale-separated in all spatial directions (i.e. where the state of non-equilibrium in the fluid can be adequately described by a single strain tensor and temperature gradient vector), our field-wise coupled HMM has no such limitations and so can be applied to flows with arbitrarily-varying degrees of scale separation (e.g. flow from a large reservoir into a nano-channel). The second major advantage is that the position of molecular elements does not need to be collocated with nodes of the continuum grid, which means that the resolution of the microscopic correction can be adjusted independently of the resolution of the continuum model. This in turn means the computational cost and accuracy of the molecular correction can be independently controlled and optimised. The macroscopic constraints on the individual molecular solvers are artificial body-force distributions, used in conjunction with standard periodicity. We test our hybrid method on the Poiseuille flow problem for both Newtonian (Lennard-Jones) and non-Newtonian (FENE) fluids. The multiscale results are validated with expensive full-scale molecular dynamics simulations of the same case. Very close agreement is obtained for all cases, with as few as two micro elements required to accurately capture both the Newtonian and non-Newtonian flowfields. Our multiscale method converges very quickly (within 3-4 iterations) and is an order of magnitude more computationally efficient than the full-scale simulation.

Switching criteria for hybrid rarefied gas flow solvers

Switching criteria for hybrid hydrodynamic/molecular gas flow solvers are developed, and are demonstrated to be more appropriate than conventional criteria for identifying thermodynamic non-equilibrium. For switching from a molecular/kinetic solver to a hydrodynamic (continuum-fluid) solver, the criterion is based on the difference between the hydrodynamic near-equilibrium fluxes (i.e. the Navier–Stokes stress and Fourier heat flux) and the actual values of stress and heat flux as computed from the molecular solver. For switching from hydrodynamics to molecular/kinetic, a similar criterion is used but the values of stress and heat flux are approximated through higher order constitutive relations; in this case, we use the R13 equations. The efficacy of our proposed switching criteria is tested within an illustrative hybrid kinetic/Navier–Stokes solver. For the test cases investigated, the results from the hybrid procedure compare very well with the full kinetic solution, and are obtained at a fraction of the computational cost.

Numerical simulation of the interaction of microactuators and boundary layers

A technique is presented for carrying out relatively low-cost numerical simulations of the interaction between three-dimensional microelectromechanical systems (MEMS)- and mesoscale actuators and a laminar boundary layer. The jet-type actuators take the form of a diaphragmlocated at the bottom of a cavity. When the diaphragm is driven by piezoceramic, for example, it de� ects, reduces the cavity volume, and drives air out of an ori� ce as a jet into the boundary layer. In an attempt to avoid an in� ow phase into the cavity, we study the effects of a “puff-like” jet produced when the diaphragmis driven by a short-duration constant force, or the cavity pressure is suddenly increased by providing air from a microvalve. The theoretical model for the actuator is based on classic thin-plate theory for the diaphragmdynamics andmodi� ed unsteady pipe-� owtheory for the � uid dynamics in the ori� ce/nozzle leading to the boundary layer. The cavity � uid dynamics is not modeled in detail; the compressible � owinit is neglected, and the instantaneouspressure there is determined viathe perfect gas law.A velocity–vorticity method is used to compute the perturbation � ow� eld created in the boundary layer. This method is capable of full direct numerical simulations, but for the present results the governing equations were linearized. The cavity and boundary-layer � ow� elds are linked by requiring continuity of velocity and pressure at the ori� ce exit. The computational methods are used to investigate such questions as the need for fully interactive computations and the differences between meso- and MEMS-scale actuators.