The Experts below are selected from a list of 198 Experts worldwide ranked by ideXlab platform
Mémin Etienne - One of the best experts on this subject based on the ideXlab platform.
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Stochastic representation of the Reynolds Transport Theorem: revisiting large-scale modeling
'Elsevier BV', 2017Co-Authors: Kadri Harouna Souleymane, Mémin EtienneAbstract:International audienceWe explore the potential of a formulation of the Navier-Stokes equations incorporating a random description of the small-scale velocity component. This model, established from a version of the Reynolds Transport Theorem adapted to a stochastic representation of the flow, gives rise to a large-scale description of the flow dynamics in which emerges an anisotropic subgrid tensor, reminiscent to the Reynolds stress tensor, together with a drift correction due to an inhomogeneous turbulence. The corresponding subgrid model, which depends on the small scales velocity variance, generalizes the Boussinesq eddy viscosity assumption. However, it is not anymore obtained from an analogy with molecular dissipation but ensues rigorously from the random modeling of the flow. This principle allows us to propose several subgrid models defined directly on the resolved flow component. We assess and compare numerically those models on a standard Green-Taylor vortex flow at Reynolds numbers Re=1600, Re=3000 and Re=5000. The numerical simulations, carried out with an accurate divergence-free scheme, outperform classical large-eddies formulations and provides a simple demonstration of the pertinence of the proposed large-scale modeling
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High-resolution data assimilation through stochastic subgrid tensor and parameter estimation from 4DEnVar
'Emerging Health Threats Forum CIC', 2017Co-Authors: Yang Yin, Mémin EtienneAbstract:International audienceIn this paper we explore a dynamical formulation allowing to assimilate high resolution data in a large-scale fluid flow model. This large-scale formulation relies on a random modeling of the small-scale velocity component and allows to take into account the scale discrepancy between the dynamics and the observations. It introduces a subgrid stress tensor that naturally emerges from a modified Reynolds Transport Theorem adapted to this stochastic representation of the flow. This principle is used within a stochastic shallow water model coupled with an 4DEnVar assimilation technique to estimate both the flow initial conditions and the inhomogeneous time-varying subgrid parameters. The performance of this modeling has been assessed numerically with both synthetic and real world data. Our strategy has shown to be very effective in providing a more relevant prior/posterior ensemble in terms of the dispersion compared to other tests using the standard shallow water equations with no subgrid parameterization or with simple eddy viscosity models. We also compared two localization techniques. The results indicate the localized covariance approach is more suitable to deal with the scale discrepancy related errors
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Stochastic parameterization of geophysical flows through modelling under location uncertainty
HAL CCSD, 2016Co-Authors: Resseguier Valentin, Mémin Etienne, Chapron Bertrand, Dérian PierreAbstract:International audienceIn this talk we will describe a framework for the systematic derivation of stochastic representations of geophysical flows. This paradigm, quoted as modelling under location uncertainty, relies on a Lagrangian decomposition of the flow velocity in terms of a large-scale, smooth in time, component and a random field uncorrelated in time, which represents the small-scale velocity component. This possibly anisotropic non-homogeneous random field corresponds to the aliasing of the unresolved velocity component. Such a Lagrangian decomposition leads to a stochastic representation of the Reynolds Transport Theorem (RTT) and of the material derivative [1,2]. Those expressions involve a diffusive subgrid term balanced by a multiplicative noise and a modified advection drift induced by the small-scale inhomogeneity. The stochastic material derivative together with the RTT enables us to express random versions any geophysical flow dynamics within the usual physical scaling approximation.Through the presentation we will provide several stochastic representations of classical systems. Quasi-Geostrophic models (QG), and Surface Quasi Geostrophic (SQG) models will be in particular explored. We will show how different SQG approximations can be derived from different levels of noise; we will demonstrate that such systems lead to improved large-scale representations and meaningful ensemble of realizations. Compared to traditional ensemble built from a perturbation of the initial condition, the ensemble generated by the proposed stochastic representation exhibits a larger spread; this allows estimating accurately the model errors in terms of location and magnitude [2]; it leads also to efficient tracking of likely scenarios [3].The nice properties of this derivation should be particularly useful for ensemble-based data assimilation techniques or for ensemble forecasting analysis. [1] E. Mémin, Fluid flow dynamics under location uncertainty, Geophysical & Astrophysical Fluid Dynamics, 108, 2, 119–146, (2014).[2] V. Resseguier, E. Mémin, B. Chapron (2016). Geophysical flows under location uncertainty, paper submitted to Geophysical & Astrophysical Fluid Dynamics.[3] V. Resseguier, E. Mémin, B. Chapron (2016). Chaotic transitions and location uncertainty in geophysical flows, paper submitted to Chaos
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Oceanic models under uncertainty
HAL CCSD, 2016Co-Authors: Resseguier Valentin, Mémin Etienne, Chapron BertrandAbstract:International audienceStochastic models can be developed to perform ensemble forecasts of geophysical fluid dynamical systems to more efficiently handle subgrid parametrizations. As numerical simulations do not usually resolve all temporal scales, solutions can be found by decomposing the velocity into a smooth resolved component and an unresolved one, uncorrelated in time and inhomogeneous in space. In turn, this new point of view changes the usual interpretation of Transport and fundamental conservation laws (mass, momentum and 1st principle). Indeed, following a stochastic version of the Reynolds Transport Theorem (Mémin, 2014), three terms naturally emerge: a multiplicative noise, an anisotropic and inhomogeneous diffusion, and a drift correction. For instance, neglecting diabatic effects implies the conservation of the temperature T along the flow as : DT/Dt=0, with the material derivative D/Dt, now understood in a stochastic sense.As such, the rigorous derivation of this model relates the random forcing to the subgrid parametrization. This further ensures important properties such as energy conservation and also simplifies the derivation of ensemble forecasts in defining a clear mathematical framework.Using this new framework, stochastic versions of geophysical models have been derived, namely: Navier-Stokes equations in a rotating frame, the Boussinesq approximation, Quasi-Geostrophy (QG) and Surface Quasi-Geostrophy (SQG). Depending on the amount of randomness, the QG approximation can lead to two different models. With moderate uncertainty, the horizontal Transport of the Potential Vorticity (PV), in the interior of the fluid, has 3 sources terms. For homogeneous turbulence, two of them, disappear. The remaining term, a noise uncorrelated in time, encodes the interactions between the resolved and the unresolved velocity gradient tensors. The ensuing SQG model is then derived assuming a zero PV in the interior, as in the deterministic case (Held et al., 1995). Yet, the buoyancy Transport at the surface of the fluid has to be understood in the stochastic sense. Increasing to strong uncertainty, the QG approximation further leads to a vanishing PV in the interior. As such, a classical SQG relationship remains. However, for that case, an ageostrophic component appears, contributing to intensify asymmetries between cyclones and anticyclones, as in the SQG+ model (Hakim et al., 2002). Frontogenesis occurs on the cold side of fronts, on uncertain locations, and frontolysis smooths the warm side of fronts.As obtained, the flow topology is changed by the noise and the divergent component, and the symmetry breaks more rapidly. Larger the uncertainty, faster the symmetry is broken. Simulating an ensemble of realizations then enables to track the different possible topologies and all the bifurcations of the system
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Stochastic Reynolds Theorem and generalized subgrid tensor
HAL CCSD, 2015Co-Authors: Resseguier Valentin, Mémin Etienne, Chapron BertrandAbstract:International audienceWe propose a representation that allows decomposing the flow velocity in terms of a smooth component and a highly oscillating random component. This decomposion leads through a stochastic representation of the Reynolds Transport Theorem to a large-scale expression of the Navier-Stokes equations. In this work we show the benefit of such a representation to construct low order dynamical systems that include naturally a dissipative term related to the action of the small-scale random component
Noack, Bernd R. - One of the best experts on this subject based on the ideXlab platform.
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Rethinking the Reynolds Transport Theorem, Liouville Equation, and Perron-Frobenius and Koopman Operators
2020Co-Authors: Niven, Robert K., Cordier Laurent, Kaiser Eurika, Schlegel Michael, Noack, Bernd R.Abstract:The Reynolds Transport Theorem provides a generalized conservation law for a conserved quantity carried by fluid flow through a continuous connected control volume. 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. In this work, new generalized formulations of these Theorems and operators are presented in different spaces. These include (a) spatial maps between different positions in a time-independent flow, 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. The findings are applied to a number of fluid mechanical and dynamical systems, including spatial (time-independent) and spatiotemporal fluid flows, flow systems with pairwise or $n$-wise spatial correlations, phase space systems, Lagrangian flows, spectral flows, and systems with coupled chemical and flow processes.Comment: 7 figure file
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Rethinking the Reynolds Transport Theorem, Liouville Equation, and Perron-Frobenius and Koopman Operators
HAL CCSD, 2019Co-Authors: Niven, Robert K., Cordier Laurent, Kaiser Eurika, Schlegel Michael, Noack, Bernd R.Abstract:7 figure filesThe 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
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Control Volume Analysis, Entropy Balance and the Entropy Production in Flow Systems
2014Co-Authors: Niven, Robert K., Noack, Bernd R.Abstract:This chapter concerns "control volume analysis", the standard engineering tool for the analysis of flow systems, and its application to entropy balance calculations. Firstly, the principles of control volume analysis are enunciated and applied to flows of conserved quantities (e.g. mass, momentum, energy) through a control volume, giving integral (Reynolds Transport Theorem) and differential forms of the conservation equations. Several definitions of steady state are discussed. The concept of "entropy" is then established using Jaynes' maximum entropy method, both in general and in equilibrium thermodynamics. The thermodynamic entropy then gives the "entropy production" concept. Equations for the entropy production are then derived for simple, integral and infinitesimal flow systems. Some technical aspects are examined, including discrete and continuum representations of volume elements, the effect of radiation, and the analysis of systems subdivided into compartments. A Reynolds decomposition of the entropy production equation then reveals an "entropy production closure problem" in fluctuating dissipative systems: even at steady state, the entropy production based on mean flow rates and gradients is not necessarily in balance with the outward entropy fluxes based on mean quantities. Finally, a direct analysis of an infinitesimal element by Jaynes' maximum entropy method yields a theoretical framework with which to predict the steady state of a flow system. This is cast in terms of a "minimum flux potential" principle, which reduces, in different circumstances, to maximum or minimum entropy production (MaxEP or MinEP) principles. It is hoped that this chapter inspires others to attain a deeper understanding and higher technical rigour in the calculation and extremisation of the entropy production in flow systems of all types.Comment: Reference: Niven, R.K. and Noack, B.R. (2013), Control volume analysis, entropy balance and the entropy production in flow systems, in Dewar R.C., Lineweaver C., Niven R.K., Regenauer-Lieb K., Beyond the Second Law: Entropy Production and Non-Equilibrium Systems, Springer-Verlag, Berlin, Heidelberg, ISBN 978-3-642-40153-4, pp 129-16
Bernd R. Noack - One of the best experts on this subject based on the ideXlab platform.
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Rethinking the Reynolds Transport Theorem, Liouville Equation, and Perron-Frobenius and Koopman Operators
2019Co-Authors: Robert K. Niven, Laurent Cordier, Eurika Kaiser, Michael Schlegel, Bernd R. NoackAbstract: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.
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rethinking the Reynolds Transport Theorem liouville equation and perron frobenius and koopman operators spatial and generalized parametric forms
arXiv: Fluid Dynamics, 2018Co-Authors: Robert K. Niven, Bernd R. Noack, Laurent Cordier, Eurika Kaiser, Michael Schlegel, Nicolas HerouardAbstract:We exploit a lesser-known connection between the (temporal) Reynolds Transport Theorem, Reynolds averaging and the Liouville equation for the flow of a conserved quantity, to derive new spatial and parametric forms of these Theorems and associated evolution operators, which provide maps between different domains (in various spaces) associated with a conserved quantity in a vector or tensor field. First, for a time-independent continuous flow field described by Eulerian velocity and position coordinates $(\u,\x)$, we derive spatial analogs of the Reynolds Transport Theorem and Liouville equation -- the latter based on the joint-conditional probability density function $f(\u | \x)$ -- and spatial analogs of the Perron-Frobenius and Koopman operators. These provide spatial maps between different positions within a velocity gradient field. For intrinsic motion (with a fixed tensorial frame of reference), the spatial mapping is induced by the shear stress tensor field. The analysis is then generalized to derive parametric Reynolds Transport Theorems and Liouville equations -- the latter based either on a probability differential form (using a generalized Lie derivative and other operators) or probability density function -- and generalized parametric Perron-Frobenius and Koopman operators. The analyses reveal the existence of multivariate Lie symmetries (in time, space or general parametric coordinates) induced by a vector or tensor field associated with the flow of a conserved quantity. The findings are illustrated by application to a variety of fluid flow and dynamical systems, including turbulent flow, two-point and $n$-point correlation, Lagrangian, phase space, spectral and chemical reaction systems.
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Control Volume Analysis, Entropy Balance and the Entropy Production in Flow Systems
Understanding Complex Systems, 2013Co-Authors: Robert K. Niven, Bernd R. NoackAbstract:This chapter concerns “control volume analysis”, the standard engineering tool for the analysis of flow systems, and its application to entropy balance calculations. Firstly, the principles of control volume analysis are enunciated and applied to flows of conserved quantities (e.g. mass, momentum, energy) through a control volume, giving integral (Reynolds Transport Theorem) and differential forms of the conservation equations. Several definitions of steady state are discussed. The concept of “entropy” is then established using Jaynes’ maximum entropy method, both in general and in equilibrium thermodynamics. The thermodynamic entropy then gives the “entropy production” concept. Equations for the entropy production are then derived for simple, integral and infinitesimal flow systems. Some technical aspects are examined, including discrete and continuum representations of volume elements, the effect of radiation, and the analysis of systems subdivided into compartments. A Reynolds decomposition of the entropy production equation then reveals an “entropy production closure problem” in fluctuating dissipative systems: even at steady state, the entropy production based on mean flow rates and gradients is not necessarily in balance with the outward entropy fluxes based on mean quantities. Finally, a direct analysis of an infinitesimal element by Jaynes’ maximum entropy method yields a theoretical framework with which to predict the steady state of a flow system. This is cast in terms of a “minimum flux potential” principle, which reduces, in different circumstances, to maximum or minimum entropy production (MaxEP or MinEP) principles. It is hoped that this chapter inspires others to attain a deeper understanding and higher technical rigour in the calculation and extremisation of the entropy production in flow systems of all types.
Robert K. Niven - One of the best experts on this subject based on the ideXlab platform.
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Rethinking the Reynolds Transport Theorem, Liouville Equation, and Perron-Frobenius and Koopman Operators
2019Co-Authors: Robert K. Niven, Laurent Cordier, Eurika Kaiser, Michael Schlegel, Bernd R. NoackAbstract: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.
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rethinking the Reynolds Transport Theorem liouville equation and perron frobenius and koopman operators spatial and generalized parametric forms
arXiv: Fluid Dynamics, 2018Co-Authors: Robert K. Niven, Bernd R. Noack, Laurent Cordier, Eurika Kaiser, Michael Schlegel, Nicolas HerouardAbstract:We exploit a lesser-known connection between the (temporal) Reynolds Transport Theorem, Reynolds averaging and the Liouville equation for the flow of a conserved quantity, to derive new spatial and parametric forms of these Theorems and associated evolution operators, which provide maps between different domains (in various spaces) associated with a conserved quantity in a vector or tensor field. First, for a time-independent continuous flow field described by Eulerian velocity and position coordinates $(\u,\x)$, we derive spatial analogs of the Reynolds Transport Theorem and Liouville equation -- the latter based on the joint-conditional probability density function $f(\u | \x)$ -- and spatial analogs of the Perron-Frobenius and Koopman operators. These provide spatial maps between different positions within a velocity gradient field. For intrinsic motion (with a fixed tensorial frame of reference), the spatial mapping is induced by the shear stress tensor field. The analysis is then generalized to derive parametric Reynolds Transport Theorems and Liouville equations -- the latter based either on a probability differential form (using a generalized Lie derivative and other operators) or probability density function -- and generalized parametric Perron-Frobenius and Koopman operators. The analyses reveal the existence of multivariate Lie symmetries (in time, space or general parametric coordinates) induced by a vector or tensor field associated with the flow of a conserved quantity. The findings are illustrated by application to a variety of fluid flow and dynamical systems, including turbulent flow, two-point and $n$-point correlation, Lagrangian, phase space, spectral and chemical reaction systems.
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Control Volume Analysis, Entropy Balance and the Entropy Production in Flow Systems
Understanding Complex Systems, 2013Co-Authors: Robert K. Niven, Bernd R. NoackAbstract:This chapter concerns “control volume analysis”, the standard engineering tool for the analysis of flow systems, and its application to entropy balance calculations. Firstly, the principles of control volume analysis are enunciated and applied to flows of conserved quantities (e.g. mass, momentum, energy) through a control volume, giving integral (Reynolds Transport Theorem) and differential forms of the conservation equations. Several definitions of steady state are discussed. The concept of “entropy” is then established using Jaynes’ maximum entropy method, both in general and in equilibrium thermodynamics. The thermodynamic entropy then gives the “entropy production” concept. Equations for the entropy production are then derived for simple, integral and infinitesimal flow systems. Some technical aspects are examined, including discrete and continuum representations of volume elements, the effect of radiation, and the analysis of systems subdivided into compartments. A Reynolds decomposition of the entropy production equation then reveals an “entropy production closure problem” in fluctuating dissipative systems: even at steady state, the entropy production based on mean flow rates and gradients is not necessarily in balance with the outward entropy fluxes based on mean quantities. Finally, a direct analysis of an infinitesimal element by Jaynes’ maximum entropy method yields a theoretical framework with which to predict the steady state of a flow system. This is cast in terms of a “minimum flux potential” principle, which reduces, in different circumstances, to maximum or minimum entropy production (MaxEP or MinEP) principles. It is hoped that this chapter inspires others to attain a deeper understanding and higher technical rigour in the calculation and extremisation of the entropy production in flow systems of all types.
Niven, Robert K. - One of the best experts on this subject based on the ideXlab platform.
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Rethinking the Reynolds Transport Theorem, Liouville Equation, and Perron-Frobenius and Koopman Operators
2020Co-Authors: Niven, Robert K., Cordier Laurent, Kaiser Eurika, Schlegel Michael, Noack, Bernd R.Abstract:The Reynolds Transport Theorem provides a generalized conservation law for a conserved quantity carried by fluid flow through a continuous connected control volume. 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. In this work, new generalized formulations of these Theorems and operators are presented in different spaces. These include (a) spatial maps between different positions in a time-independent flow, 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. The findings are applied to a number of fluid mechanical and dynamical systems, including spatial (time-independent) and spatiotemporal fluid flows, flow systems with pairwise or $n$-wise spatial correlations, phase space systems, Lagrangian flows, spectral flows, and systems with coupled chemical and flow processes.Comment: 7 figure file
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Rethinking the Reynolds Transport Theorem, Liouville Equation, and Perron-Frobenius and Koopman Operators
HAL CCSD, 2019Co-Authors: Niven, Robert K., Cordier Laurent, Kaiser Eurika, Schlegel Michael, Noack, Bernd R.Abstract:7 figure filesThe 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
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Control Volume Analysis, Entropy Balance and the Entropy Production in Flow Systems
2014Co-Authors: Niven, Robert K., Noack, Bernd R.Abstract:This chapter concerns "control volume analysis", the standard engineering tool for the analysis of flow systems, and its application to entropy balance calculations. Firstly, the principles of control volume analysis are enunciated and applied to flows of conserved quantities (e.g. mass, momentum, energy) through a control volume, giving integral (Reynolds Transport Theorem) and differential forms of the conservation equations. Several definitions of steady state are discussed. The concept of "entropy" is then established using Jaynes' maximum entropy method, both in general and in equilibrium thermodynamics. The thermodynamic entropy then gives the "entropy production" concept. Equations for the entropy production are then derived for simple, integral and infinitesimal flow systems. Some technical aspects are examined, including discrete and continuum representations of volume elements, the effect of radiation, and the analysis of systems subdivided into compartments. A Reynolds decomposition of the entropy production equation then reveals an "entropy production closure problem" in fluctuating dissipative systems: even at steady state, the entropy production based on mean flow rates and gradients is not necessarily in balance with the outward entropy fluxes based on mean quantities. Finally, a direct analysis of an infinitesimal element by Jaynes' maximum entropy method yields a theoretical framework with which to predict the steady state of a flow system. This is cast in terms of a "minimum flux potential" principle, which reduces, in different circumstances, to maximum or minimum entropy production (MaxEP or MinEP) principles. It is hoped that this chapter inspires others to attain a deeper understanding and higher technical rigour in the calculation and extremisation of the entropy production in flow systems of all types.Comment: Reference: Niven, R.K. and Noack, B.R. (2013), Control volume analysis, entropy balance and the entropy production in flow systems, in Dewar R.C., Lineweaver C., Niven R.K., Regenauer-Lieb K., Beyond the Second Law: Entropy Production and Non-Equilibrium Systems, Springer-Verlag, Berlin, Heidelberg, ISBN 978-3-642-40153-4, pp 129-16