Importance Sampling Variance Reduction for the Fokker-Planck Rarefied Gas Particle Method

Models and methods that are able to accurately and efficiently predict the flows of low-speed rarefied gases are in high demand, due to the increasing ability to manufacture devices at micro and nano scales. One such model and method is a Fokker-Planck approximation to the Boltzmann equation, which can be solved numerically by a stochastic particle method. The stochastic nature of this method leads to noisy estimates of the thermodynamic quantities one wishes to sample when the signal is small in comparison to the thermal velocity of the gas. Recently, Gorji et al have proposed a method which is able to greatly reduce the variance of the estimators, by creating a correlated stochastic process which acts as a control variate for the noisy estimates. However, there are potential difficulties involved when the geometry of the problem is complex, as the method requires the density to be solved for independently. Importance sampling is a variance reduction technique that has already been shown to successfully reduce the noise in direct simulation Monte Carlo calculations. In this paper we propose an importance sampling method for the Fokker-Planck stochastic particle scheme. The method requires minimal change to the original algorithm, and dramatically reduces the variance of the estimates. We test the importance sampling scheme on a homogeneous relaxation, planar Couette flow and a lid-driven-cavity flow, and find that our method is able to greatly reduce the noise of estimated quantities. Significantly, we find that as the characteristic speed of the flow decreases, the variance of the noisy estimators becomes independent of the characteristic speed

On the modelling of isothermal gas flows at the microscale

This paper makes two new propositions regarding the modelling of rarefied (non-equilibrium) 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

Coupled continuum hydrodynamics and molecular dynamics method for multiscale simulation

We present a new hybrid methodology for carrying out multiscale simulations of flow problems lying between continuum hydrodynamics and molecular dynamics, where macro/micro lengthscale separation exists only in one direction. Our multiscale method consists of an iterative technique that couples mass and momentum flux between macro and micro domains, and is tested on a converging/diverging nanochannel case containing flow of a simple Lennard-Jones liquid. Comparisons agree well with a full MD simulation of the same test case

Dynamics of liquid nano-threads : fluctuation-driven instability and rupture

The instability and rupture of nanoscale liquid threads is shown to strongly depend on thermal fluctuations. These fluctuations are naturally occurring within molecular dynamics (MD) simulations and can be incorporated via fluctuating hydrodynamics into a stochastic lubrication equation (SLE). A simple and robust numerical scheme is developed for the SLE that is validated against MD for both the initial (linear) instability and the nonlinear rupture process. Particular attention is paid to the rupture process and its statistics, where the `double-cone’ profile reported by Moseler & Landmann [Science, 2000, 289(5482): 1165-1169] is observed, as well as other distinct profile forms depending on the flow conditions. Comparison to the Eggers’ similarity solution [Physical Review Letters, 2002, 89(8): 084502], a power law of the minimum thread radius against time to rupture, shows agreement only at low surface tension; indicating that surface tension cannot generally be neglected when considering rupture dynamics

Numerical simulation of boundary-layer control using MEMS actuation

MEMS actuators and their effect on boundary layers is investigated using numerical simulation. The thesis is specifically focussed on jet actuators and their application to the targeted control of turbulent boundary layers. A complete numerical model of jet-type actuators, including the popular synthetic-jet actuator, is developed. The assumed input is the voltage signal to the piezocermic driver and the calculated output is the exit jet velocity. Thorough validation of the numerical code is presented and simulations performed to highlight the key issues in MEMS-actuator design. The three-dimensional boundary-layer disturbance created by the MEMS actuator is modelled using a velocity-vorticity formulation of the Navier-Stokes equations. The parallel code is rigorously validated against results from linear stability theory and transitional-streak measurements. The boundary-layer code is used to determine a performance criterion for MEMS jets; it is shown that the net mass flow from a jet best determines its effectiveness. The code is also used to demonstrate the macro-scale capabilities of MEMS-scale actuators; a grid-scaling method is described and employed to facilitate this calculation. A method is presented that enables high- and low-speed streaks to be modelled economically in otherwise undisturbed mean flows. Using this model, the fundamental principles of targeted control using MEMS actuation are explored. The MEMS-actuator and boundary-layer models are coupled, and an investigation into the interactive effects of the two systems is described. Using the coupled code, disturbances in the boundary layer are shown to induce velocities in inactive devices. One special case occurs when an oscillating pressure field creates Helmhotz resonance within the cavity of a MEMS actuator, thus causing large mass flow rates in and out of the device. It is also suggested that the MEMS device could strongly interact with the random fluctuations of a turbulent boundary layer, leading to highly unpredictable actuator responses

A wall-function approach to incorporating Knudsen-layer effects in gas micro flow simulations

For gas flows in microfluidic configurations, the Knudsen layer close to the wall can comprise a substantial part of the entire flow field and has a major effect on quantities such as the mass flow rate through micro devices. The Knudsen layer itself is characterized by a highly nonlinear relationship between the viscous stress and the strain rate of the gas, so even if the Navier-Stokes equations can be used to describe the core gas flow they are certainly inappropriate for the Knudsen layer itself. In this paper we propose a "wall-function" model for the stress/strain rate relations in the Knudsen layer. The constitutive structure of the Knudsen layer has been derived from results from kinetic theory for isothermal shear flow over a planar surface. We investigate the ability of this simplified model to predict Knudsen-layer effects in a variety of configurations. We further propose a semi-empirical Knudsen-number correction to this wall function, based on high-accuracy DSMC results, to extend the predictive capabilities of the model to greater degrees of rarefaction

Capturing the Knudsen layer in continuum-fluid models of non-equilibrium 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

Multiscale simulation of non-isothermal microchannel gas flows

AbstractThis paper describes the development and application of an efficient hybrid continuum-molecular approach for simulating non-isothermal, low-speed, internal rarefied gas flows, and its application to flows in Knudsen compressors. The method is an extension of the hybrid continuum-molecular approach presented by Patronis et al. (2013) [4], which is based on the framework originally proposed by Borg et al. (2013) [3] for the simulation of micro/nano flows of high aspect ratio. The extensions are: 1) the ability to simulate non-isothermal flows; 2) the ability to simulate low-speed flows by implementing a molecular description of the gas provided by the low-variance deviational simulation Monte Carlo (LVDSMC) method; and 3) the application to three-dimensional geometries. For the purposes of validation, the multiscale method is applied to rarefied gas flow through a periodic converging-diverging channel (driven by an external acceleration). For this flow problem it is computationally feasible to obtain a solution by the direct simulation Monte Carlo (DSMC) method for comparison: very close agreement is observed.The efficiency of the multiscale method, allows the investigation of alternative Knudsen-compressor channel configurations to be undertaken. We characterise the effectiveness of the single-stage Knudsen-compressor channel by the pressure drop that can be achieved between two connected reservoirs, for a given temperature difference. Our multiscale simulations indicate that the efficiency is surprisingly robust to modifications in streamwise variations of both temperature and cross-sectional geometry

A generalised optimisation principle for asymmetric branching in fluidic networks

When applied to a branching network, Murray’s law states that the optimal branching of vascular networks is achieved when the cube of the parent channel radius is equal to the sum of the cubes of the daughter channel radii. It is considered integral to understanding biological networks and for the biomimetic design of artificial fluidic systems. However, despite its ubiquity, we demonstrate that Murray’s law is only optimal (i.e. maximises flow conductance per unit volume) for symmetric branching, where the local optimisation of each individual channel corresponds to the global optimum of the network as a whole. In this paper, we present a generalised law that is valid for asymmetric branching, for any cross-sectional shape, and for a range of fluidic models. We verify our analytical solutions with the numerical optimisation of a bifurcating fluidic network for the examples of laminar, turbulent, and non-Newtonian fluid flows