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

Dynamics and stability of lean-premixed swirl-stabilized combustion

A hierarchy of low-dimensional Galerkin models is proposed for the viscous, incompressible flow around a circular cylinder building on the pioneering works of Stuart (1958), Deane et al. (1991), and Ma & Karniadakis (2002). The empirical Galerkin model is based on an eight-dimensional Karhunen–Loeve decomposition of a numerical simulation and incorporates a new ‘shift-mode’ representing the mean-field correction. The inclusion of the shift-mode significantly improves the resolution of the transient dynamics from the onset of vortex shedding to the periodic von Karman vortex street. In addition, the Reynolds-number dependence of the flow can be described with good accuracy. The inclusion of stability eigenmodes further enhances the accuracy of fluctuation dynamics. Mathematical and physical system reduction approaches lead to invariant-manifold and to mean-field models, respectively. The corresponding two-dimensional dynamical systems are further reduced to the Landau amplitude equation.

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A hierarchy of low-dimensional models for the transient and post-transient cylinder wake

/pdf/model-reduction-for-compressible-flows-using-pod-and-54usfx7h91.pdf

Model reduction for compressible flows using POD and Galerkin projection

Progress in probability density function methods for turbulent reacting flows

An investigation is presented that analyses the energy flows that are connected to the dynamical behaviour of coherent structures in a transitional flat-plate boundary layer. Based on a mathematical description of the three-dimensional coherent structures of this flow as provided by the Karhunen–Loeve procedure, energy equations for the coherent structures are derived by Galerkin projection of the Navier–Stokes equations in vorticity transport formulation onto the corresponding basis of eigenfunctions. In a first step, the time-averaged energy balance – showing the energy flows that support the different coherent structures and thus maintain the fluctuations of the velocity field – is considered. In a second step, the instantaneous power budget is investigated for the particularly interesting case of a coherent structure providing a prime contribution to the characteristic spike events of the transitional boundary layer. As this structure shows a strong variation in energy, the question about which mechanisms cause these variations is addressed. Our results show that the occurrence of a spike must be attributed to an autonomous event and cannot be interpreted as just an epiphenomenon of the passage of a Λ-vortex.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112094003551

Evolution of three-dimensional coherent structures in a flat-plate boundary layer

In this paper some implications of the technique of projecting the Navier–Stokes equations onto low-dimensional bases of eigenfunctions are explored. Such low-dimensional bases are typically obtained by truncating a particularly well-suited complete set of eigenfunctions at very low orders, arguing that a small number of such eigenmodes already captures a large part of the dynamics of the system. In addition, in the treatment of inhomogeneous spatial directions of a flow, eigenfunctions that do not satisfy the boundary conditions are often used, and in the Galerkin projection the corresponding boundary conditions are ignored. We show how the restriction to a low-dimensional basis as well as improper treatment of boundary conditions can affect the range of validity of these models. As particular examples of eigenfunction bases, systems of Karhunen–Loeve eigenfunctions are discussed in more detail, although the results presented are valid for any basis.

On Low-Dimensional Galerkin Models for Fluid Flow

The proper orthogonal decomposition (POD) is applied to both computational fluid dynamics and particle imaging velocimetry data of simplified motored engine flows using two different methods. The first method is to apply the POD to ensembles of velocity fields obtained by considering the flow field taken at fixed crank-angle positions over a number of cycles. As a result, sets of POD modes are found, each of which describe the structure of the flow at a given piston position. These sets give some indication of the instability mechanism involved in tumble breakdown. The second method we use represents a novel approach of applying the POD to flows within a time-varying domain. The velocity fields are stretched to a fixed domain and normalized so that all phases of the flow are equally weighted. In this way, ‘phase-invariant POD modes’ are created. The phase-invariant modes show desirable properties for forming a suitable basis for future low-dimensional models which should describe the breakdown process mor...

Dietmar Rempfer

Papers

Evolution of three-dimensional coherent structures in a flat-plate boundary layer

On Low-Dimensional Galerkin Models for Fluid Flow

Application of the proper orthogonal decomposition to datasets of internal combustion engine flows

Numerical study of detailed flow field and performance of Savonius wind turbines

On Boundary Conditions for Incompressible Navier-Stokes Problems