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1. Lagrangian Averaged Stochastic Advection by Lie Transport for Fluids

   https://doi.org/10.1007/s10955-020-02493-4  

   Drivas, Theodore D. ; Holm, Darryl D. ; Leahy, James-Michael ( June 2020 , Journal of Statistical Physics)

   null (Ed.)

   Abstract We formulate a class of stochastic partial differential equations based on Kelvin’s circulation theorem for ideal fluids. In these models, the velocity field is randomly transported by white-noise vector fields, as well as by its own average over realizations of this noise. We call these systems the Lagrangian averaged stochastic advection by Lie transport (LA SALT) equations. These equations are nonlinear and non-local, in both physical and probability space. Before taking this average, the equations recover the Stochastic Advection by Lie Transport (SALT) fluid equations introduced by Holm (Proc R Soc A 471(2176):20140963, 2015). Remarkably, the introduction of the non-locality in probability space in the form of momentum transported by its own mean velocity gives rise to a closed equation for the expectation field which comprises Navier–Stokes equations with Lie–Laplacian ‘dissipation’. As such, this form of non-locality provides a regularization mechanism. The formalism we develop is closely connected to the stochastic Weber velocity framework of Constantin and Iyer (Commun Pure Appl Math 61(3):330–345, 2008) in the case when the noise correlates are taken to be the constant basis vectors in $$\mathbb {R}^3$$ R 3 and, thus, the Lie–Laplacian reduces to the usual Laplacian. We extend this class of equations to allow for advected quantities to be present and affect the flow through exchange of kinetic and potential energies. The statistics of the solutions for the LA SALT fluid equations are found to be changing dynamically due to an array of intricate correlations among the physical variables. The statistical properties of the LA SALT physical variables propagate as local evolutionary equations which when spatially integrated become dynamical equations for the variances of the fluctuations. Essentially, the LA SALT theory is a non-equilibrium stochastic linear response theory for fluctuations in SALT fluids with advected quantities. 

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2. Embedding properties of network realizations of dissipative reduced order models

   Druskin, Guddati ( July 2022 , HOUSEHOLDER SYMPOSIUM XXI)

   Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and block-tridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistor-capacitor-inductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to so-called nite-dierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM state-space representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from the transfer function without solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse problems and unsupervised machine learning. Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic wave propagation in lossy dispersive media. Potential applications include solution of inverse scattering problems in dispersive media, such as seismic exploration, radars and sonars. To x the idea, we consider a passive irreducible SISO ROM fn(s) = Xn j=1 yi s + σj , (62) assuming that all complex terms in (62) come in conjugate pairs. We will seek ladder realization of (62) as rjuj + vj − vj−1 = −shˆjuj , uj+1 − uj + ˆrj vj = −shj vj , (63) for j = 0, . . . , n with boundary conditions un+1 = 0, v1 = −1, and 4n real parameters hi, hˆi, ri and rˆi, i = 1, . . . , n, that can be considered, respectively, as the equivalent discrete inductances, capacitors and also primary and dual conductors. Alternatively, they can be viewed as respectively masses, spring stiness, primary and dual dampers of a mechanical string. Reordering variables would bring (63) into tridiagonal form, so from the spectral measure given by (62 ) the coecients of (63) can be obtained via a non-symmetric Lanczos algorithm written in J-symmetric form and fn(s) can be equivalently computed as fn(s) = u1. The cases considered in the original FDGQR correspond to either (i) real y, θ or (ii) real y and imaginary θ. Both cases are covered by the Stieltjes theorem, that yields in case (i) real positive h, hˆ and trivial r, rˆ, and in case (ii) real positive h,r and trivial hˆ,rˆ. This result allowed us a simple interpretation of (62) as the staggered nite-dierence approximation of the underlying PDE problem [2]. For PDEs in more than one variables (including topologically rich data-manifolds), a nite-dierence interpretation is obtained via a MIMO extensions in block form, e.g., [4, 3]. The main diculty of extending this approach to general passive problems is that the Stieltjes theory is no longer applicable. Moreover, the tridiagonal realization of a passive ROM transfer function (62) via the ladder network (63) cannot always be obtained in port-Hamiltonian form, i.e., the equivalent primary and dual conductors may change sign [1]. 100 Embedding of the Stieltjes problems, e.g., the case (i) was done by mapping h and hˆ into values of acoustic (or electromagnetic) impedance at grid cells, that required a special coordinate stretching (known as travel time coordinate transform) for continuous problems. Likewise, to circumvent possible non-positivity of conductors for the non-Stieltjes case, we introduce an additional complex s-dependent coordinate stretching, vanishing as s → ∞ [1]. This stretching applied in the discrete setting induces a diagonal factorization, removes oscillating coecients, and leads to an accurate embedding for moderate variations of the coecients of the continuum problems, i.e., it maps discrete coecients onto the values of their continuum counterparts. Not only does this embedding yields an approximate linear algebraic algorithm for the solution of the inverse problems for dissipative PDEs, it also leads to new insight into the properties of their ROM realizations. We will also discuss another approach to embedding, based on Krein-Nudelman theory [5], that results in special data-driven adaptive grids. References [1] Borcea, Liliana and Druskin, Vladimir and Zimmerling, Jörn, A reduced order model approach to inverse scattering in lossy layered media, Journal of Scientic Computing, V. 89, N1, pp. 136,2021 [2] Druskin, Vladimir and Knizhnerman, Leonid, Gaussian spectral rules for the three-point second dierences: I. A two-point positive denite problem in a semi-innite domain, SIAM Journal on Numerical Analysis, V. 37, N 2, pp.403422, 1999 [3] Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Distance preserving model order reduction of graph-Laplacians and cluster analysis, Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Journal of Scientic Computing, V. 90, N 1, pp 130, 2022 [4] Druskin, Vladimir and Moskow, Shari and Zaslavsky, Mikhail LippmannSchwingerLanczos algorithm for inverse scattering problems, Inverse Problems, V. 37, N. 7, 2021, [5] Mark Adolfovich Nudelman The Krein String and Characteristic Functions of Maximal Dissipative Operators, Journal of Mathematical Sciences, 2004, V 124, pp 49184934 Go back to Plenary Speakers Go back to Speakers Go back 

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3. Reciprocal swimming at intermediate Reynolds number

   https://doi.org/10.1017/jfm.2022.873  

   Derr, Nicholas J. ; Dombrowski, Thomas ; Rycroft, Chris H. ; Klotsa, Daphne ( December 2022 , Journal of Fluid Mechanics)

   In Stokes flow, Purcell's scallop theorem forbids objects with time-reversible (reciprocal) swimming strokes from moving. In the presence of inertia, this restriction is eased and reciprocally deforming bodies can swim. A number of recent works have investigated dimer models that swim reciprocally at intermediate Reynolds numbers ${\textit Re} \approx 1$ –1000. These show interesting results (e.g. switches of the swim direction as a function of inertia) but the results vary and seem to be case specific. Here, we introduce a general model and investigate the behaviour of an asymmetric spherical dimer of oscillating length for small-amplitude motion at intermediate ${\textit {Re}}$ . In our analysis we make the important distinction between particle and fluid inertia, both of which need to be considered separately. We asymptotically expand the Navier–Stokes equations in the small-amplitude limit to obtain a system of linear partial differential equations. Using a combination of numerical (finite element) and analytical (reciprocal theorem, method of reflections) methods we solve the system to obtain the dimer's swim speed and show that there are two mechanisms that give rise to motion: boundary conditions (an effective slip velocity) and Reynolds stresses. Each mechanism is driven by two classes of sphere–sphere interactions, between one sphere's motion and (1) the oscillating background flow induced by the other's motion, and (2) a geometric asymmetry induced by the other's presence. We can thus unify and explain behaviours observed in other works. Our results show how sensitive, counterintuitive and rich motility is in the parameter space of finite inertia of particles and fluid. 

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4. Performance Optimization and Exergy Analysis of Thermoelectric Heat Recovery System for Gas Turbine Power Plants

   https://doi.org/10.3390/e25121583  

   Alsaghir, Ahmad M. ; Bahk, Je-Hyeong ( December 2023 , Entropy)

   Thermoelectric (TE) waste heat recovery has attracted significant attention over the past decades, owing to its direct heat-to-electricity conversion capability and reliable operation. However, methods for application-specific, system-level TE design have not been thoroughly investigated. This work provides detailed design optimization strategies and exergy analysis for TE waste heat recovery systems. To this end, we propose the use of TE system equipped on the exhaust of a gas turbine power plant for exhaust waste heat recovery and use it as a case study. A numerical tool has been developed to solve the coupled charge and heat current equations with temperature-dependent material properties and convective heat transfer at the interfaces with the exhaust gases at the hot side and with the ambient air at the heat sink side. Our calculations show that at the optimum design with 50% fill factor and 6 mm leg thickness made of state-of-the-art Bi2Te3 alloys, the proposed system can reach power output of 10.5 kW for the TE system attached on a 2 m-long, 0.5 × 0.5 m2-area exhaust duct with system efficiency of 5% and material cost per power of 0.23 $/W. Our extensive exergy analysis reveals that only 1% of the exergy content of the exhaust gas is exploited in this heat recovery process and the exergy efficiency of the TE system can reach 8% with improvement potential of 85%.

    

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5. Flux-based modeling of heat and mass transfer in multicomponent systems

   https://doi.org/10.1063/5.0085444  

   Beris, Antony N. ; Jariwala, Soham ; Wagner, Norman J. ( March 2022 , Physics of Fluids)

   In the present work, the macroscopic governing equations governing the heat and mass transfer for a general multicomponent system are derived via a systematic nonequilibrium thermodynamics framework. In contrast to previous approaches, the relative (with respect to the mass average velocity) component mass fluxes (relative species momenta) and the heat flux are treated explicitly, in complete analogy with the momentum flux. The framework followed here, in addition to allowing for the description of relaxation phenomena in heat and mass transfer, establishes to the fullest the analogy between all transport processes, momentum, heat, and mass transfer, toward which R. B. Bird contributed so much with his work. The inclusion of heat flux-based momentum as an additional variable allows for the description of relaxation phenomena in heat transfer as well as of mixed (Soret and Dufour) effects, coupling heat and mass transfer. The resulting models are Galilean invariant, thereby resolving a conundrum in the field, and always respect the second law of thermodynamics, for appropriate selection of transport parameters. The general flux-based dynamic equations reduce to the traditional transport equations in the limit when mass species and heat relaxation effects are negligible and are fully consistent with the equations established from the application of kinetic theory in the limit of dilute gases. As an added benefit, for the particular example case of hyperbolic diffusion we illustrate the application of the proposed models as a method to allow the use of powerful numerical solvers normally not available for solving mass transfer models more generally.

    

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