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

The numerical simulation of high Reynolds number flows is hampered by model accuracy if the Reynolds-averaged Navier–Stokes (RANS) equations are used, and by computational cost if direct or large-eddy simulations (LES) that resolve the near-wall layer are employed. The cost of a calculation scales like the Reynolds number to the power 3 for direct numerical simulations, or 2.4 for LES, making the resolution of the wall layer at high Reynolds number infeasible even with the most advanced computers. In LES, an attractive alternative to compute high-Re flows is the use of wall-layer models, in which only the outer layer is resolved, while the near-wall region is modeled. Three broad classes of approaches are presently used: bypassing this region altogether using wall functions, solving a separate set of equations in the near-wall region, weakly coupled to the outer flow, or simulating the near-wall region in a global, Reynolds-averaged, sense. These approaches are discussed and their ranges of applicability are highlighted. Various unresolved issues in wall-layer modeling are presented.

Wall-layer models for large-eddy simulations

The Structure of Turbulent Shear Flow By Dr. A. A. Townsend. Pp. xii + 315. 8¾ in. × 5½ in. (Cambridge: At the University Press.) 40s.

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The Structure of Turbulent Shear Flow

We analyse the currently popular vortex identification criteria that are based on point-wise analysis of the velocity gradient tensor. A new measure of spiralling compactness of material orbits in vortices is introduced and using this measure a new local vortex identification criterion and requirements for a vortex core are proposed. The inter-relationships between the different criteria are explored analytically and in a few flow examples, using both zero and non-zero thresholds for the identification parameter. These inter-relationships provide a new interpretation of the various criteria in terms of the local flow kinematics. A canonical turbulent flow example is studied, and it is observed that all the criteria, given the proposed usage of threshold, result in remarkably similar looking vortical structures. A unified interpretation based on local flow kinematics is offered for when similarity or differences can be expected in the vortical structures educed using the different criteria.

On the relationships between local vortex identification schemes

The identification issue of coherent vortices is investigated on the basis of direct numerical simulation (DNS) and large-eddy simulations (LES) of turbulent flows. It is first shown that the pressure Laplacian is positive within a low-pressure tube of small cross section enclosed by convex isobaric surfaces in a uniform-density flow. Since this quantity is related to the second invariant Q of ∇ u , the Q criterion (region where Q is positive) is a necessary condition for the existence of such tubes. This eduction scheme is compared to other classical methods in incompressible simulations of isotropic turbulence: a mixing layer, a channel flow and a backward-facing step. Q-isosurfaces turn out to display very nice coherent vortices. This criterion is also used in combination with a conditional sampling method to discuss the characteristics of quasi-longitudinal vortices in a manipulated channel flow. The contribution of near-wall vortical structures to velocity and vorticity fluctuations is clearly isolat...

On coherent-vortex identification in turbulence

Numerical simulations of turbulent polymer solutions using the FENE-P model are used to characterize the action of polymers on turbulence in drag-reduced flows. The energetics of turbulence is investigated by correlating the work done by polymers on the flow with turbulent structures. Polymers are found to store and to release energy to the flow in a well-organized manner. The storage of energy occurs around near-wall vortices as has been anticipated for a long time. Quite unexpectedly, coherent release of energy is observed in the very near-wall region. Large fluctuations of polymer work are shown to re-energize decaying streamwise velocity fluctuations in high-speed streaks just above the viscous sublayer. These distinct behaviours are used to propose an autonomous regeneration cycle of polymer wall turbulence, in the spirit of Jimenez & Pinelli (1999).

On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows

Turbulence is ubiquitous in nature, yet even for the case of ordinary Newtonian fluids like water, our understanding of this phenomenon is limited Many liquids of practical importance are more complicated (eg, blood, polymer melts, paints), however; they exhibit elastic as well as viscous characteristics, and the relation between stress and strain is nonlinear We demonstrate here for a model system of such complex fluids that at high shear rates, turbulence is not simply modified as previously believed but is suppressed and replaced by a different type of disordered motion, elasto-inertial turbulence Elasto-inertial turbulence is found to occur at much lower Reynolds numbers than Newtonian turbulence, and the dynamical properties differ significantly The friction scaling observed coincides with the so-called “maximum drag reduction” asymptote, which is exhibited by a wide range of viscoelastic fluids

https://www.pnas.org/content/pnas/110/26/10557.full.pdf

Elasto-inertial turbulence.

Elasto-inertial turbulence (EIT) is a new state of turbulence found in inertial flows with polymer additives. The dynamics of turbulence generated and controlled by such additives is investigated from the perspective of the coupling between polymer dynamics and flow structures. Direct numerical simulations of channel flow with Reynolds numbers ranging from 1000 to 6000 (based on the bulk and the channel height) are used to study the formation and dynamics of elastic instabilities and their effects on the flow. The flow topology of EIT is found to differ significantly from Newtonian wall-turbulence. Structures identified by positive (rotational flow topology) and negative (extensional/compressional flow topology) second invariant Qa isosurfaces of the velocity gradient are cylindrical and aligned in the spanwise direction. Polymers are significantly stretched in sheet-like regions that extend in the streamwise direction with a small upward tilt. The Qa cylindrical structures emerge from the sheets of high polymer extension, in a mechanism of energy transfer from the fluctuations of the polymer stress work to the turbulent kinetic energy. At subcritical Reynolds numbers, EIT is observed at modest Weissenberg number (Wi, ratio polymer relaxation time to viscous time scale). For supercritical Reynolds numbers, flows approach EIT at large Wi. EIT provides new insights on the nature of the asymptotic state of polymer drag reduction (maximum drag reduction), and explains the phenomenon of early turbulence, or onset of turbulence at lower Reynolds numbers than for Newtonian flows observed in some polymeric flows.

On the mechanism of elasto-inertial turbulence

Numerical data of polymer drag reduced flows is interpreted in terms of modification of near-wall coherent structures. The originality of the method is based on numerical experiments in which boundary conditions or the governing equations are modified in a controlled manner to isolate certain features of the interaction between polymers and turbulence. As a result, polymers are shown to reduce drag by damping near-wall vortices and sustain turbulence by injecting energy onto the streamwise velocity component in the very near-wall region.

Yves Dubief

Papers

On coherent-vortex identification in turbulence

On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows

Elasto-inertial turbulence.

On the mechanism of elasto-inertial turbulence

New Answers on the Interaction Between Polymers and Vortices in Turbulent Flows