Michael Schlegel

A Finite-Time Thermodynamics of Unsteady Fluid Flows

Abstract Turbulent fluid has often been conceptualized as a transient thermodynamic phase. Here, a finite-time thermodynamics (FTT) formalism is proposed to compute mean flow and fluctuation levels of unsteady incompressible flows. The proposed formalism builds upon the Galerkin model framework, which simplifies a continuum 3D fluid motion into a finite-dimensional phase-space dynamics and, subsequently, into a thermodynamics energy problem. The Galerkin model consists of a velocity field expansion in terms of flow configuration dependent modes and of a dynamical system describing the temporal evolution of the mode coefficients. Each mode is treated as one thermodynamic degree of freedom, characterized by an energy level. The dynamical system approaches local thermal equilibrium (LTE) where each mode has the same energy if it is governed only by internal (triadic) mode interactions. However, in the generic case of unsteady flows, the full system approaches only partial LTE with unequal energy levels due to strongly mode-dependent external interactions. The FTT model is first illustrated by a traveling wave governed by a 1D Burgers equation. It is then applied to two flow benchmarks: the relatively simple laminar vortex shedding, which is dominated by two eigenmodes, and the homogeneous shear turbulence, which has been modeled with 1459 modes.

Identifying noisy and quiet modes in a jet

In the current jet noise study, an empirical modal decomposition is proposed which distills the noisy and quiet modes of the flow field. In particular, the POD of flows is generalised for an optimal resolution of the far-field noise as opposed to a least-order representation of the hydrodynamic fluctuation level. This decomposition technique, which we call ‘most observable decomposition (MOD)’, is based on a linear cause-eect relationship between the hydrodynamics (cause) and the far-field acoustics (observed eect). In the current study, this relationship is identified from a linear stochastic estimation between the flow field and the far-field pressure — taking into account the propagation time of sound. We employ MOD to turbulent jet noise at Ma = 0.9, Re = 3600 using CFD/CAA data from RWTH Aachen. While more than 350 POD modes are necessary to capture only 50% of the flow fluctuation energy, a mere 24 MOD modes resolve 90% of the far-field acoustics. Evidently, far-field noise acts as filter which ‘sees’ only a low-dimensional subspace of the flow and ‘ignores’ silent subspaces which contain a large amount of fluctuation energy. The MOD methodology yields ‘least-order’ representations of any other observable as well — assuming a linear relationship between flow and observable.

Coherent Structures in a Transitional Flow around a Backward‐Facing Step

The transitional flow around a backward-facing step visualized using illuminated streamlines of a snapshot ~Fig. 1!. The flow separates at the corner of the step. The resu shear layer rolls up in two Kelvin–Helmholtz vortices. In th downstream direction, the streamlines form bundles due secondary streamwise vorticity. The fluid experiences a sm backward flow in the upstream region below the shear la The flow field is obtained from a large-eddy simulation Kaltenbach and Janke at a Reynolds number of Re H53000 based on oncoming velocity and on step height. The co sponding boundary conditions are described in Ref. 1. The streamlines are illuminated in order to enhance

Galerkin Method for Nonlinear Dynamics

A Galerkin method is presented for control-oriented reduced-order models (ROM). This method generalizes linear approaches elaborated by M. Morzynski et al. for the nonlinear Navier-Stokes equation. These ROM are used as plants for control design in the chapters by G. Tadmor et al., S. Siegel, and R. King in this volume. Focus is placed on empirical ROM which compress flow data in the proper orthogonal decomposition (POD). The chapter shall provide a complete description for construction of straight-forward ROM as well as the physical understanding and teste

Reduced-Order Modelling of Turbulent Jets for Noise Control

A reduced-ordermodelling (ROM) strategy is pursued to achieve a mechanistic understanding of jet flow mechanisms targeting jet noise control. Coherent flow structures of the jet are identified by the proper orthogonal decomposition (POD) and wavelet analysis. These techniques are applied to an LES data ensemble with velocity snapshots of a three-dimensional, incompressible jet at a Reynolds number of Re=3600. A low-dimensionalGalerkin model of a three-dimensional jet is extracted and calibrated to the physical dynamics. To obtain the desired mechanistic understanding of jet noise generation, the loudest flow structures are distilled by a goal-oriented generalisation of the POD approach we term ’most observable decomposition’ (MOD). Thus, a reduction of the number of dynamically most important degrees of freedom by one order of magnitude is achieved. Capability of the presented ROM strategy for jet noise control is demonstrated by suppression of loud flow structures.

Turbulence Control Based on Reduced-Order Models and Nonlinear Control Design

We present a closed-loop flow control strategy for experiments and simulations. This strategy is based on low-order Galerkin models and nonlinear control. One key enabler is a partitioning of the flow in low-, dominant- and high-frequency components, i.e. a base flow, coherent structures and stochastic fluctuations. Another enabler is a control design exploiting the nonlinearities distilled by the model. Examples are presented for the actuated flow around a high-lift configuration and the controlled bluff body wake.

Maximum entropy analysis of hydraulic pipe networks

A Maximum Entropy (MaxEnt) method is developed to infer mean external and internal flow rates and mean pressure gradients (potential differences) in hydraulic pipe networks, without or with sufficient constraints to render the system deterministic. The proposed method substantially extends existing methods for the analysis of flow networks (e.g. Hardy-Cross), applicable only to deterministic networks.

Maximum entropy analysis of flow networks

This study examines a generalised maximum entropy (MaxEnt) analysis of a flow network, involving flow rates and potential differences on the network, connected by resistance functions. The analysis gives a generic derivation based on an explicit form of the resistance functions. Accounting for the constraints also leads to an extended form of Gibbs’ phase rule, applicable to flow networks.

Maximum entropy analysis of flow and reaction networks

We present a generalised MaxEnt method to infer the stationary state of a flow network, subject to “observable” constraints on expectations of various parameters, as well as “physical” constraints arising from frictional properties (resistance functions) and conservation laws (Kirchhoff laws). The method invokes an entropy defined over all uncertainties in the system, in this case the internal and external flow rates and potential differences. The proposed MaxEnt framework is readily extendable to the analysis of networks with uncertainty in the network structure itself.

Reduced-Parameter Method for Maximum Entropy Analysis of Hydraulic Pipe Flow Networks

AbstractA maximum entropy (MaxEnt) method is developed to predict flow rates or pressure gradients in hydraulic pipe networks without sufficient information to give a closed-form (deterministic) so...