The Dirichlet-to-Neumann operator on rough domains with finite volume

A consistent dynamic localization model for large eddy simulation of turbulent flows based on a variational formulation

We show the David-Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require its surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As a consequence every Wolff snowflake has infinite surface measure.

Semisolid sets and topological measures

Error estimation and control are critical ingredients for improving the reliability of computational simulations. Adjoint-based techniques can be used to both estimate the error in chosen solution outputs and to provide local indicators for adaptive refinement. This article reviews recent work on these techniques for computational fluid dynamics applications in aerospace engineering. The definition of the adjoint as the sensitivity of an output to residual source perturbations is used to derive both the adjoint equation, in fully discrete and variational formulations, and the adjoint-weighted residual method for error estimation. Assumptions and approximations made in the calculations are discussed. Presentation of the discrete and variational formulations enables a side-by-side comparison of recent work in output-error estimation using the finite volume method and the finite element method. Techniques for adapting meshes using output-error indicators are also reviewed. Recent adaptive results from a variety of laminar and Reynolds-averaged Navier-Stokes applications show the power of output-based adaptive methods for improving the robustness of computational fluid dynamics computations. However, challenges and areas of additional future research remain, including computable error bounds and robust mesh adaptation mechanics.

Box schemes (finite volume methods) are widely used in fluiddynamics, especially for the solution of conservation laws. In this paper two box-schemes for elliptic equations are analysed with respect to quadrilateral meshes. Using a variational formulation, we gain stability theorems for two different box methods, namely the so-called diagonal boxes and the centre boxes. The analysis is based on an elementwise eigenvalue problem. Stability can only be guaranteed under additional assumptions on the geometry of the quadrilaterals. For the diagonal boxes unsuitable elements can lead to global instabilities. The centre boxes are more robust and differ not so much from the finite element approach. In the stable case, convergence results up to second order are proved with well-known techniques.

Fins are commonly employed for cooling of electronic equipment, compressors, Internal Combustion engines and for heat exchange in various heat exchangers. In short fin (length to height ratio, L/H = 5) arrays used for natural convection cooling, a stagnation zone forms at the central portion and that portion is not effective for carrying away heat. An attempt is made to modify plate fin heat sink geometry (PFHS) by inserting pin fins in the channels formed between plate fins and a plate fin pin fin heat sink (PFPFHS) is constructed to address this issue. An experimental setup is developed to validate numerical model of PFPFHS. The three-dimensional elliptic governing equations were solved using a finite volume based computational fluid dynamics (CFD) code. Fluent 6.3.26, a finite volume flow solver is used for solving the set of governing equations for the present geometry. Cell count based on grid independence and extended domain is used to obtain numerical results. Initially, the numerical model is validated for PFHS cases reported in the literature. After obtaining a good agreement with results from the literature, the numerical model for PFHS is modified for PFPFHS and used to carry out systematic parametric study of PFPFHS to analyze the effects of parameters like fin spacing, fin height, pin fin diameter, number of pin fins and temperature difference between fin array and surroundings on natural convection heat transfer from PFPFHS. It is observed that it is impossible to obtain optimum performance in terms of overall heat transfer by only concentrating on one or two parameters. The interactions among all the design parameters must be considered. This thesis presents Experimental and Numerical study of natural convection heat transfer from horizontal rectangular plate fin and plate fin pin fin arrays. The parameters of study are fin spacing, temperature difference between the fin surface and ambient air, fin height, pin fin diameter, number of pin fins and method of positioning pin fins in the fin channel. Experimental set up is validated with horizontal plate standard correlations. Results are generated in the form of variation in average heat transfer coefficient (ha), base heat transfer coefficient (hb), average Nusselt number (Nua) and base Nusselt number (Nub). Total 512 cases are studied numerically and finally an attempt is made to correlate the Nusselt Number (Nu), Rayleigh Number (Ra), increase in percentage by inserting pin fins (% Area), ratios like spacing to height (S/H) and L/H obtained in the present study.

We prove that an open set $\Omega \subset \mathbb{R}^n$ can be approximated by smooth sets of uniformly bounded perimeter from the interior if and only if the open set $\Omega$ satisfies \begin{align*} &\qquad \qquad\qquad\qquad\qquad\qquad\qquad \mathscr{H}^{n-1}(\partial \Omega \setminus \Omega^0)<\infty, \qquad &&\quad\qquad\qquad \qquad\qquad (*) \end{align*} where $\Omega^0$ is the measure-theoretic exterior of $\Omega$. Furthermore, we show that condition (*) implies that the open set $\Omega$ is an extension domain for bounded divergence-measure fields, which improves the previous results that require a strong condition that $\mathscr{H}^{n-1}(\partial \Omega)<\infty$. As an application, we establish a Gauss-Green formula up to the boundary on any open set $\Omega$ satisfying condition (*) for bounded divergence-measure fields, for which the corresponding normal trace is shown to be a bounded function concentrated on $\partial \Omega \setminus \Omega^0$. This new formula does not require the set of integration to be compactly contained in the domain where the vector field is defined. In addition, we also analyze the solvability of the divergence equation on a rough domain with prescribed trace on the boundary, as well as the extension domains for bounded $BV$ functions.

We provide an N/V-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on ℝ d ,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝ d with finite volume (Lebesgue measure) V=|Λ|<∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above N-particle dynamic in Λ as N→∞ and V→∞ such that N/V→ρ, where ρ is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate relation between N and V. Then tightness is shown by using the Lyons–Zheng decomposition. The equilibrium measures of the accumulation points are identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle, infinite volume stochastic dynamics via the associated martingale problem. Assuming a property closely related to Markov uniqueness and weaker than essential self-adjointness, via Mosco convergence techniques we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials ϕ of Ruelle type and all temperatures, densities, and dimensions d≥1, respectively. ϕ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides as a by-product an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary condition.

We study the problem of continuum percolation in infinite volume Gibbs measures for particles with an attractive pair potential, with a focus on low temperatures (large β). The main results are bounds on percolation thresholds ρ ± (β) in terms of the density rather than the chemical potential or activity. In addition, we prove a variational formula for a large deviations rate function for cluster size distributions. This formula establishes a link with the Gibbs variational principle and a form of equivalence of ensembles, and allows us to combine knowledge on finite volume, canonical Gibbs measures with infinite volume, grand-canonical Gibbs measures

A new variable extrapolation formulation for unstructured finite volume codes is developed which closely resembles the MUSCL-scheme used within structured flow solvers. This new formulation is based on information currently available to the unstructured flow solvers, namely the variable information and the gradient information, and as such, it is trivial to implement within most finite volume flow solvers. This new variable extrapolation formulation represents a one-parameter family of equations and under certain circumstances, it is fully equivalent to the MUSCL-scheme, which is also a one-parameter family. A wide variety of results are presented, including theoretical analysis of the truncation error and numerical analysis of the truncation error for a one-dimensional problem, using the method of manufactured solutions, as well as for a two-dimensional problem. Inviscid three-dimensional results are presented which demonstrate that the numerical viscosity can be greatly decreased via this formulation, and the viscous results for a variety of cases indicate that the drag is better predicted and that the vortical structures within the flow field are captured more accurately and further downstream of their origination. Finally, the observed improved convergence levels and stability of the new formulation are discussed, as well as the issues involved with extending this approach to achieve third-order spatial accuracy for 2D and 3D grids.

Estimating the state of charge in a latent thermal energy storage heat exchanger based on inlet/outlet and surface measurements

Formation and development of a breaker bar under regular waves. Part 1: Model description and hydrodynamics

A computational model is developed to predict the performance of phase change materials(PCMs) for passive thermal control of electronic modules during transient power variations or following an active cooling system failure. Two different ways of incorporating PCM in the module are considered. One is to place a laminate of PCM outside the multichip module, and the other is to place the PCM laminate between the substrate and the cold plate. Two different types of PCMs are considered. One is n-Eicosene, which is an organic paraffin, and the other one is a eutectic alloy of Bi/Pb/Sn/In. Computations are performed in three dimensions using a finite volume method. A single domain fixed grid enthalpy porosity method is used to model the effects of phase change. Effects of natural convection on the performance of PCM are also examined. Results are presented in the form of time-wise variations of maximum module temperature, isotherm contours, velocity vectors, and melt front locations. Effects of PCM laminate thickness and power levels are studied to assess the amount of PCM required for a particular power level. The results show that the PCMs are an effective option for passive cooling of high density electronic modules for transient periods.

Nowadays, limited energy resources face ever-growing demands of the modern world. One engineering approach to mitigate this problem which has received considerable attention in recent years is using latent heat thermal storage (LHTS) systems, a significant opportunity which is provided by phase change materials (PCMs). In the present study, a numerical investigation was devoted to estimate the simultaneous freezing and melting processes of a double-layer PCM in terms of heat transfer and fluid flow phenomena. A double-pipe cylindrical channel with two compartments, A and B, was considered for locating two PCMs of RT28 and RT35 in various arrangements. The inner and outer walls were exposed to both hot and cold heat transfer fluids (HHTFs and CHTFs, respectively) beginning with solid or liquid initial state, which led to solid–liquid phase change process through PCMs. The numerical simulation was handled by a two-dimensional finite volume method (FVM) with a fixed Rayleigh number of 106 in which conduction and convection heat transfer mechanisms are taken into account. The effects of employing double-layer PCM and their arrangements, inner and outer walls’ boundary conditions, and initial statuses of PCMs are discussed, and the details of the compared results are shown in the form of temperature and liquid fraction variations over time.

The effect of radiation on natural convection in slanted cavities of angle γ = 45° and 60°