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 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.

We investigate linear-quadratic dynamical systems with energy-preserving quadratic terms. These systems arise for instance as Galerkin systems of incompressible flows. A criterion is presented to ensure long-term boundedness of the system dynamics. If the criterion is violated, a globally stable attractor cannot exist for an effective nonlinearity. Thus, the criterion can be considered a minimum requirement for control-oriented Galerkin models of viscous fluid flows. The criterion is exemplified, for example, for Galerkin systems of two-dimensional cylinder wake flow models in the transient and the post-transient regime, for the Lorenz system and for wall-bounded shear flows. There are numerous potential applications of the criterion, for instance, system reduction and control of strongly nonlinear dynamical systems.

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-effect relationship between the hydrodynamics (cause) and the far-field acoustics (observed effect). 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 M a = 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 wellassuming a linear relationship between flow and observable.

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SUMMARYWe propose a system reduction strategy for spectral and Galerkin models of incompressible fluid flows. This approach leads to dynamic models of lower order, based on a partition in slow, dominant and fast modes. In the reduced models, slow dynamics are incorporated as non-linear manifold consistent with mean-field theory. Fast dynamics are stochastically treated and can be lumped in eddyviscosity approaches. The employed interaction models between slow, dominant and fast dynamics respect momentum and energy balance equations in a mathematically rigorous manner-unlike unsteady Reynoldsaveraged Navier-Stokes models or Smagorinsky-type reductions of the Navier-Stokes equation. The proposed system reduction strategy is employed to the cylinder wake benchmark.

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The Reynolds transport theorem provides a generalised conservation law for any conserved quantity carried by fluid flow through a continuous domain, and underpins all integral and differential analyses of flow systems. It is also intimately linked to the Liouville equation for the conservation of a local probability density function (pdf), and to the Perron-Frobenius and Koopman evolution operators. All of these tools can be interpreted as continuous temporal maps between fluid elements or domains, connected by the integral curves (pathlines) described by a velocity vector field. We present new formulations of these theorems and operators in different spaces. These include (a) spatial maps between different positions in a time-independent flow field, connected by a velocity gradient tensor field, and (b) parametric maps -expressed using an extended exterior calculus -between different positions in a manifold, connected by a vector or tensor field. The analyses reveal the existence of multivariate continuous (Lie) symmetries induced by a vector or tensor field associated with a conserved quantity, which will be manifested in all subsidiary conservation laws such as the Navier-Stokes and energy equations. The analyses significantly expand the scope of methods for the reduction of fluid flow and dynamical systems.

We propose a generalization of proper orthogonal decomposition (POD) for optimal flow resolution of linearly related observables. This Galerkin expansion, termed 'observable inferred decomposition' (OID), addresses a need in aerodynamic and aeroacoustic applications by identifying the modes contributing most to these observables. Thus, OID constitutes a building block for physical understanding, leastbiased conditional sampling, state estimation and control design. From a continuum of OID versions, two variants are tailored for purposes of observer and control design, respectively. Firstly, the most probable flow state consistent with the observable is constructed by a 'least-residual' variant. This version constitutes a simple, easily generalizable reconstruction of the most probable hydrodynamic state to preprocess efficient observer design. Secondly, the 'least-energetic' variant identifies modes with the largest gain for the observable. This version is a building block for Lyapunov control design. The efficient dimension reduction of OID as compared to POD is demonstrated for several shear flows. In particular, three aerodynamic and aeroacoustic goal functionals are studied: (i) lift and drag fluctuation of a two-dimensional cylinder wake flow; (ii) aeroacoustic density fluctuations measured by a sensor array and emitted from a two-dimensional compressible mixing layer; † Email address for correspondence: michael.schlegel@tu-berlin.de