Jan Östh

On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body

We investigate a hierarchy of eddy-viscosity terms in proper orthogonal decomposition (POD) Galerkin models to account for a large fraction of unresolved fluctuation energy. These Galerkin methods are applied to large eddy simulation (LES) data for a flow around a vehicle-like bluff body called an Ahmed body. This flow has three challenges for any reduced-order model: a high Reynolds number, coherent structures with broadband frequency dynamics, and meta-stable asymmetric base flow states. The Galerkin models are found to be most accurate with modal eddy viscosities as proposed by Rempfer & Fasel (J. Fluid Mech., vol. 260, 1994a, pp. 351–375; J. Fluid Mech. vol. 275, 1994b, pp. 257–283). Robustness of the model solution with respect to initial conditions, eddy-viscosity values and model order is achieved only for state-dependent eddy viscosities as proposed by Noack, Morzynski & Tadmor (Reduced-Order Modelling for Flow Control, CISM Courses and Lectures, vol. 528, 2011). Only the POD system with state-dependent modal eddy viscosities can address all challenges of the flow characteristics. All parameters are analytically derived from the Navier–Stokes-based balance equations with the available data. We arrive at simple general guidelines for robust and accurate POD models which can be expected to hold for a large class of turbulent flows.

Cluster-based reduced-order modelling of a mixing layer

We propose a novel cluster-based reduced-order modelling (CROM) strategy of unsteady flows. CROM combines the cluster analysis pioneered in Gunzburgers group (Burkardt et al. 2006) and and transition matrix models introduced in fluid dynamics in Eckhardts group (Schneider et al. 2007). CROM constitutes a potential alternative to POD models and generalises the Ulam-Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron-Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Secondly, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows

We propose a variational approach to the identification of an optimal nonlinear eddy viscosity as a subscale turbulence representation for proper orthogonal decomposition (POD) models. The ansatz for the eddy viscosity is given in terms of an arbitrary function of the resolved fluctuation energy. This function is found as a minimizer of a cost functional measuring the difference between the target data coming from a resolved direct or large-eddy simulation of the flow and its reconstruction based on the POD model. The optimization is performed with a data-assimilation approach generalizing the 4D-VAR method. POD models with optimal eddy viscosities are presented for a 2D incompressible mixing layer at Re=500 (based on the initial vorticity thickness and the velocity of the high-speed stream) and a 3D Ahmed body wake at Re=300000 (based on the body height and the free-stream velocity). The variational optimization formulation elucidates a number of interesting physical insights concerning the eddy-viscosity ansatz used. The 20-dimensional model of the mixing-layer reveals a negative eddy-viscosity regime at low fluctuation levels which improves the transient times towards the attractor. The 100-dimensional wake model yields more accurate energy distributions as compared to the nonlinear modal eddy-viscosity benchmark proposed recently by Osth et al. (J. Fluid Mech., vol. 747, 2014, pp. 518–544). Our methodology can be applied to construct quite arbitrary closure relations and, more generally, constitutive relations optimizing statistical properties of a broad class of reduced-order models.

LES study of breakdown control of A-pillar vortex

Active flow control of the longitudinal vortices that developed near and around simplified A-pillar of a generic vehicle was studied using large eddy simulation (LES). The LES results were validated against existing Particle Image Velocimetry (PIV) and aerodynamic drag data. The LES results were further used to study the flow physics responsible for the development of longitudinal vortices, in particular the vortex breakdown process. Tangential blowing and suction into the shear layer rolling into the longitudinal vortices was found to be a sensitive process that can cause instabilities in the flow. The resulting LES flows also show that actuation influences not only the longitudinal vortex nearest to the actuation slot but also the overall flow. Thus, the influence of the flow control actuation on the entire flow must be considered in order to be able to find the appropriate level of control for optimal aerodynamic performance.

Cluster-based reduced-order modelling of shear flows

Cluster-based reduced-order modelling (CROM) builds on the pioneering works of Gunzburgers group in cluster analysis [1] and Eckhardts group in transition matrix models [2] and constitutes a potential alternative to reduced-order models based on a proper-orthogonal decomposition (POD). This strategy frames a time-resolved sequence of flow snapshots into a Markov model for the probabilities of cluster transitions. The information content of the Markov model is assessed with a Kullback-Leibler entropy. This entropy clearly discriminates between prediction times in which the initial conditions can be inferred by backward integration and the predictability horizon after which all information about the initial condition is lost. This approach is exemplified for a class of fluid dynamical benchmark problems like the periodic cylinder wake, the spatially evolving incompressible mixing layer, the bi-modal bluff body wake, and turbulent jet noise. For these examples, CROM is shown to distil nontrivial quasi-attr...

A LES Study of a Simplified Tractor-Trailer Model

Large-eddy simulation (LES) was used to study the flow around a simplified tractor-trailer model. The model consists of two boxes placed in tandem. The front box represents the cab of a tractor-trailer road vehicle and the rear box represents the trailer. The LES was made at the Reynolds number of \(0.51 \times 10^6\) based on the height of the rear box and the inlet air velocity. Two variants of the model were studied, one where the leading edges on the front box are sharp and one where the edges are rounded. One small and one large gap width between the two boxes were studied for both variants. Two computational grids were used in the LES simulations and a comparison was made with available experimental force measurements. The results of the LES simulations were used to analyze the flow field around the cab and in the gap between the two boxes of the tractor-trailer model. Large vortical structures around the front box and in the gap were identified. The flow field analysis showed how these large vortical structures are responsible for the difference in the drag force for the model that arises when the leading edges on the front box are rounded and the gap width is varied.

Visualizing vortex clusters in the wake of a high-speed train

Visualization of fluid flows at a high-Reynolds number (Re ∼ 105) presents difficulties for user comprehension due to density and ambiguous interactions between vortices. Prior work has used cluster-based reduced-order modelling (CROM) to analyze the wake of a High-Speed Train (HST) with Re = 86,000. In this paper, we present a novel surface visualization to convey the spatiotemporal changes undergone by clustered vortices in the HST wake. This visualization is accomplished through dimensional reduction of 3D volumetric vortices into 1D ridges, and physics-based feature tracking. The result is 3D surfaces visualizing the behavior of the vortices in the HST wake. Compared to conventional still-image representations, these surfaces allow the user to quickly compare and analyze the two shedding cycles identified via CROM. The spatiotemporal differences of the primary vortices in these shedding cycles provide analytic insight to influence the aerodynamics of the HST.