The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

This paper reviews basic results and recent developments in the field of small-scale gaseous hydrodynamics which has received significant attention in connection with small-scale science and technology. We focus on the modeling challenges arising from the breakdown of the Navier-Stokes description, observed when characteristic lengthscales become of the order of, or smaller than, the molecular mean free path. We discuss both theoretical results and numerical methods development. Examples of the former include the limit of applicability of the Navier-Stokes constitutive laws, the concept of second-order slip and the appropriate form of such a model, and how to reconcile experimental measurements of slipping flows with theory. We also review a number of recently developed theoretical descriptions of canonical nanoscale flows of engineering interest. On the simulation front, we review recent progress in characterizing the accuracy of the prevalent Boltzmann simulation method known as direct simulation Monte Carlo. We also present recent variance reduction ideas which address the prohibitive cost associated with the statistical sampling of macroscopic properties in low-speed flows.

http://absimage.aps.org/image/DFD05/MWS_DFD05-2005-020061.pdf

The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics

Flow-through assay reaction chamber (6) of cassette has back and forth liquid mixing in narrow gap (G) over array of capture agent (S), with net flow advance to waste confinement (19), produced by reversible pumps (3 or 12), operable with rolling diaphragm action with at least limited elastic recovery that advance sample or buffer liquids through conditioning paths (4A, 8, 8', 9, 14, 15, 15') before reaching the reaction chamber (6). A single pump produces accurate flow control, liquid conditioning, e.g., liquefying dry reagent from internal surfaces of flow-dividing material (14a, 15A, 15A', e.g. open cell foam or frit), heating (4A), and air bubble removal (8, 8', 9), as well as replenishment of reagent while accomplishing mixing within the flow-through reaction chamber (6). Lower viscosity buffer liquid is arranged to propel higher viscosity reagent, the flow-dividing storage material preserving reagent concentration. A blister pack (11) acts as a reversible pump (12) in producing accurate forward and backward flows with the net flow advance. Cascaded bubble traps (8, 9) on the cassette render the system tolerant of minor pumping error during cassette priming.

Assays based on liquid flow over arrays

Numerical investigation of thermal enhancement in a micro heat sink with fan-shaped reentrant cavities and internal ribs

Heat transfer in the microchannels with fan-shaped reentrant cavities and different ribs based on field synergy principle and entropy generation analysis

The next-generation of small satellites ('nanosats') will feature masses <10 kg and require miniaturised propulsion systems capable of providing extremely low levels of thrust The emergence of viscous, thermal and/or rarefaction effects on the micro-scale can significantly impact the flow behaviour in supersonic micronozzles resulting in performance characteristics which differ substantially from traditional macro-scale nozzle designs In this paper, we provide an overview of key findings obtained from computational studies of supersonic micronozzle flow and discuss the implications for future micro-scale nozzle design and optimisation

Design considerations for supersonic micronozzles

A comprehensive numerical investigation of a steady viscous flow through a two-dimensional supersonic linear micronozzle has been performed. The baseline model for the study is derived from the NASAGoddard Space Flight Center microelectromechanical systems-based hydrogen peroxide prototype microthruster. On the microscale, substantial viscous subsonic layers may form on the nozzle expander walls, which reduce thrust and efficiency. One approach to compensate for the presence of these layers has been to designmicronozzleswith expansion angles larger than traditional macroscale nozzles. Numerical simulations have been conducted for a range of Reynolds numbers (Re 15–800) and for expander half-angles of 10–50 deg. Twodifferentmonopropellant fuels have been considered: decomposed 85% pure hydrogen peroxide and decomposed hydrazine. It is found that an inherent tradeoff exists between the viscous losses and the losses resulting from the nonaxial exit flow at larger expansion angles. Our simulations indicate that themaximumnozzle efficiency occurs for bothmonopropellants at a nozzle expansion halfangle of approximately 30 deg, which is significantly larger than that of traditional conical nozzle designs.

Viscous Effects on Performance of Two-Dimensional Supersonic Linear Micronozzles

A 2-D numerical study of chaotic flow in a natural convection loop

We revisit the problem of the stability of pulsatile pipe flow for axisymmetric perturbations. In contrast to the earlier approach based on the Chebyshev expansion for the spatial discretization [J. Appl. Mech. ASME 53 (1986) 187], we use the set of the eigenfunctions derived from the longwave limit of the Orr–Sommerfeld equation. We show that the Orr–Sommerfeld basis gives greater accuracy than the Chebyshev basis if fewer terms are used in the Galerkin expansion. For the time evolution of the flow perturbation, instead of the usual Floquet analysis, a different representation for the solution of the periodic system of linear differential equations is employed. We found that the flow structures corresponding to the largest energy growth are toroidal vortex tubes. They are stretched by the shear stress of the mean flow so that a maximum energy growth occurs. The flow perturbation subsequently decays due to viscous effects. The maximum energy growth is then evaluated over a range of Reynolds and Womersley numbers. Asymptotic solutions provided for the longwave limit as well as the limit of large Womersley numbers agree well with the numerical results, confirming the known linear stability of the flow.

Darren L. Hitt

Papers

Design considerations for supersonic micronozzles

Viscous Effects on Performance of Two-Dimensional Supersonic Linear Micronozzles

A 2-D numerical study of chaotic flow in a natural convection loop

Revisiting the stability of pulsatile pipe flow

Revisiting the stability of pulsatile pipe flow