A simple and stable scale-similarity model for large Eddy simulation: Energy balance and existence of weak solutions

In averaging the Navier-Stokes equations, the problem of closure arises. Scale-similarity models address closure by (roughly speaking) extrapolation from the (known) resolved scales to the (unknown) unresolved scales. In a posteriori tests, scale-similarity models are often the most accurate but can prove to be unstable when used in a numerical simulation. In this report, we consider the scale-similarity model given by

. We prove it is stable (solutions satisfy an energy inequality) and deduce from that the existence of weak solutions of the model.     

Computation of turbulent reactive flows in industrial burners

This paper presents models that are suitable for computing steady and unsteady gaseous combustion with finite rate chemistry. Reynold averaging and large eddy simulation (LES) techniques are used to model turbulence for the steady and unsteady cases, respectively. In LES, the Reynold stress terms are modelled by a linear combination of the scale-similarity and eddy dissipation models while the cross terms are of the scale-similarity type. In Reynold averaging, the conventional k–ε two-equation model is used. For the chemical reactions, a 3-step mechanism is used for methane oxidation and the extended Zeldovich and N₂O mechanism are used for NO formation. The combustion model is a hybrid model of the Arrhenius type and a modified eddy dissipation model to take into account the effects of reaction rate, flame stretch and turbulent intensity and scale. Numerical simulations of a flat pulse burner and a swirling burner are discussed.     

Periodic Solutions of the Abelian Higgs Model and Rigid Rotation of Vortices

The metric and potential energy on the reduced moduli space of selfdual vortices in the Abelian Higgs model on are computed in a certain limit, first identified by Bradlow. In this limit it is proved that the Higgs field is asymptotic to a standard holomorphic section. These results are then used to prove a theorem asserting the existence of time-periodic solutions of the Abelian Higgs model on which represent two vortices in rigid rotation about one another. The theorem answers affirmatively the question, raised by Jaffe and Taubes, of whether a balance between the inter-vortex attraction and the centrifugal repulsion provides for the existence of such solutions (as it does in the classical two body problem for point particles.) The starting point of the analysis is the adiabatic limit system, i.e. the Hamiltonian system defined by restricting the Abelian Higgs model to the moduli space of self-dual vortices. The Hamiltonian consists of a potential energy term and kinetic energy term which is given by the metric on the moduli space induced from . It is shown under two assumptions on the metric and potential energy that the adiabatic limit system admits periodic solutions of the required type. Periodic solutions to the full system are then obtained by an application of the implicit function theorem. Explicit examples where the assumptions on the adiabatic limit system hold are provided by the computations of the metric and potential in the Bradlow limit. Submitted: March 1998.     

Fisher information and kinetic energy functionals: A dequantization approach

We strengthen the connection between information theory and quantum-mechanical systems using a recently developed dequantization procedure whereby quantum fluctuations latent in the quantum momentum are suppressed. The dequantization procedure results in a decomposition of the quantum kinetic energy as the sum of a classical term and a purely quantum term. The purely quantum term, which results from the quantum fluctuations, is essentially identical to the Fisher information. The classical term is complementary to the Fisher information and, in this sense, it plays a role analogous to that of the Shannon entropy. We demonstrate the kinetic energy decomposition for both stationary and nonstationary states and employ it to shed light on the nature of kinetic energy functionals.     

A fractional order derivative based active contour model for inhomogeneous image segmentation

Segmenting intensity inhomogeneous images is a challenging task for both local and global methods. Some hybrid methods have great advantages over the traditional methods in inhomogeneous image segmentation. In this paper, a new hybrid method is presented, which incorporates image gradient, local environment and global information into a framework, called adaptive-weighting active contour model. The energy or level set functions in the framework mainly include two parts: a global term and local term. The global term aims to enhance the image contrast, and it can also accelerate the convergence rate when minimizing the energy function. The local term integrates fractional order differentiation, fractional order gradient magnitude, and difference image information into the well-known local Chan–Vese model, which has been shown to be effective and efficient in modeling the local information. The local term can also enhance low frequency information and improve the inhomogeneous image segmentation. An adaptive weighting strategy is proposed to balance the actions of the global and local terms automatically. When minimizing the level set functions, regularization can be imposed by applying Gaussian filtering to ensure smoothness in the evolution process. In addition, a corresponding stopping criterion is proposed to ensure the evolving curve automatically stops on true boundaries of objects. Dice similarity coefficient is employed as the comparative quantitative measures for the segmented results. Experiments on synthetic images as well as real images are performed to demonstrate the segmentation accuracy and computational efficiency of the presented hybrid method.     

The LANS-alpha turbulence model in primitive-equation ocean modeling

The Lagrangian-Averaged Navier-Stokes alpha (LANS-α) and Leray turbulence parameterizations are demonstrated in a primitive-equation ocean model using an idealized channel domain. For LANS-α, turbulence statistics such as kinetic energy, eddy kinetic energy, and temperature profiles resemble doubled-resolution statistics with the standard model. In a North Atlantic domain with realistic topography, the Leray model increases eddy activity. The LANS-α and Leray models show great promise to improve heat transport and temperature distributions in global ocean-climate simulations, as these processes depend on better resolution of eddies near the grid-scale. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Improving shock-free compressible RANS solvers for LES on unstructured meshes

Standard numerical methods used to solve the Reynolds averaged Navier–Stokes equations are known to be too dissipative to carry out large eddy simulations since the artificial dissipation they introduce to stabilize the discretization of the convection term usually interacts strongly with the subgrid scale model. A possible solution is to resort to non-dissipative central schemes. Unfortunately, these schemes are in general unstable. A way to reach stability is to select a central scheme that conserves the discrete kinetic energy. To that purpose, a family of kinetic energy conserving schemes is developed to perform simulations of compressible shock-free flows on unstructured grids. A direct numerical simulation of the flow past a sphere at a Reynolds number of 300 and a large eddy simulation at a Reynolds number of 10,000 are performed to validate the methodology.     

表示湍流场的一种新设想

本文仿照量子场论中描述基本粒子产生湮灭的方法来描述湍流中涡旋的产生和消灭.因为当某一基本粒子存在的时候,我们可以认为它是一个不变实体,而湍流中涡旋则在时间过程中不断变化和耗散,所以在类比应用量子场论方法时首先要解决怎样的湍流涡旋可认为是同一个涡旋.根据线性化理论的特点,我们认为在时间过程中按相似性规律变化时湍流涡旋才算是同一个涡旋,而把不具有相似性的涡旋出现或消失,看成是方程(2.6)中相互作用项φ_i所引起的湮火和产生的结果.然后,我们采用和量子场论相类似的产生算符和消灭算符来描述湍流涡旋系统所处的状态.最后,我们利用原N-S方程中相互作用项来构成涡旋相互作用的“Schr?dinger”方程以描述其状态的变化.这样就得类似于量子场论的湍流涡旋相互作用理论.     

A similarity theory of approximate deconvolution models of turbulence

We apply the phenomenology of homogeneous, isotropic turbulence to the family of approximate deconvolution models proposed by Stolz and Adams. In particular, we establish that the models themselves have an energy cascade with two asymptotically different inertial ranges. Delineation of these gives insight into the resolution requirements of using approximate deconvolution models. The approximate deconvolution model's energy balance contains both an enhanced energy dissipation and a modification to the model's kinetic energy. The modification of the model's kinetic energy induces a secondary energy cascade which accelerates scale truncation. The enhanced energy dissipation completes the scale truncation by reducing the model's micro-scale from the Kolmogorov micro-scale.     

Energy dissipation bounds for hear flows for a model in large eddy simulation

This report gives an upper bound for the time average of the energy dissipation rate, including energy dissipation due to viscous dissipation and turbulent diffusion in a model for the motion of large eddies in a turbulent flow. The bound is only a function of the reference velocity U, the domain diameter L, the eddy size δ, and surprisingly, the Reynolds number.     

Parallelizable and robust image segmentation model based on the shape prior information

Image segmentation is an important task in many fields, and there are plentiful models based on region or edges. Nowadays, the speed of calculation and the universal applicability of the model attract much attention. To some extent, the traditional energy model can segment images suffering from intensity inhomogeneity while it relies on initialization seriously. In this paper, we present a new model that consists of an arbitrary active contour model and proposed shape priori information term, which can segment various images accurately and provide an opportunity to carry on parallelizable calculation. The shape priori information term plays a key role in our energy functional and the shape priori information can be chosen diversely. This term also improves the robustness of our model including initial conditions and parameter adjustment. Besides, the split Bregman method is then applied to minimize the energy functional. Multiple experimental results and comparisons are shown to demonstrate the superiority of the proposed model. Firstly, fuzzy clustering, threshold and manual operation are used to be the shape priori information. Secondly, it is illustrated that our model is not sensitive to parameters and initial contours. Computation time and accuracy are also obviously improved when using the parallel algorithm.     

On the blowing up of solutions to quantum hydrodynamic models on bounded domains

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the multi-dimensional quantum hydrodynamic model in a bounded domain is proved. The model consists on conservation of mass equation and a momentum balance equation equivalent to a compressible Euler equations corrected by a dispersion term of the third order in the momentum balance. The proof is based on a priori estimates for the energy functional for a new observable constructed with an auxiliary function, and it is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time.     

Fully discrete energy stable scheme for a phase-field moving contact line model with variable densities and viscosities

In this study, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model comprises a Cahn–Hilliard equation, Navier–Stokes equation, and the generalized Navier boundary condition for the moving contact line. A scalar auxiliary variable is employed to transform the governing system into an equivalent form, thereby allowing the double well potential to be treated semi-explicitly. A stabilization term is added to balance the explicit nonlinear term originating from the surface energy at the fluid–solid interface. A pressure stabilization method is used to decouple the velocity and pressure computations. Some subtle implicit–explicit treatments are employed to deal with convention and stress terms. We establish a rigorous proof of the energy stability for the proposed time-marching scheme. A finite difference method based on staggered grids is then used to spatially discretize the constructed time-marching scheme. We also prove that the fully discrete scheme satisfies the discrete energy dissipation law. Our numerical results demonstrate the accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze the contact line dynamics based on a shear flow-driven droplet sliding case. Three-dimensional droplet spreading is also investigated based on a chemically patterned surface. Our numerical simulation accurately predicts the expected energy evolution and it successfully reproduces the expected phenomena where an oil droplet contracts inward on a hydrophobic zone and then spreads outward rapidly on a hydrophilic zone.     

A mathematical model for induction hardening including mechanical effects

We investigate a mathematical model for induction hardening of steel. It accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical effects caused by the phase transitions. We take care of effects like transformation strain and transformation plasticity induced by the phase transitions and allow for physical parameters depending on the respective phase volume fractions.The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L¹ data. We prove existence of a weak solution to the complete system using a truncation argument.     

On Meeting Energy Balance Errors in Co-Simulations

The term co-simulation denotes the coupling of some simulation tools for dynamical systems into one big system by having them exchange data at points of a fixed time grid and extrapolating the received data into the interval, while none of the steps is repeated for iteration. From the global perspective, the simulation thus has a strong explicit component. Frequently, among the data passed across subsystem boundaries there are flows of conserved quantities, and as there is no iteration of steps, system-wide balances may not be fulfilled: the system is not solved as one monolithic equation system. If these balance errors accumulate, simulation results become inaccurate. Balance correction methods which compensate these errors by adding corrections for the balances to the signal in the next coupling time step have been considered in past research. But establishing the balance of one quantity a posteriori due to the time delay in general cannot establish the balances of quantities that depend on the exchanged quantities, usually energy. In most applications from physics, the balance of energy is equivalent to stability. In this paper, a method is presented which allows users to choose the quantity that should be balanced to be that energy, and to accurately balance it. This establishes also numerical stability for many classes of stable problems.     

湍流的Markov过程理论与Kolmogoroff理论的联系以及对制Kolmogoroff定律的推广——Ⅰ.对两个理论关系的分析

本文全文分为两部分(Ⅰ和Ⅱ).第Ⅰ部分讨论了关于大雷诺数湍流的两种理论——拉格朗日观点的Markov过程理论与欧拉观点的Kolmogoroff理论之间的联系.指出:对位置和速度的联合过程进行Markov描述所需雷诺数与Kolmogoroff第二相似性假设所需雷诺数同样大;周期与T_u和T_f同阶的旋涡分别对应于Kolmogoroff理论的含能范围与耗损范围;Richarson定律的适用范围T_*≤t≤β-1对应于Kolmogoroff理论的惯性子范围,从而指出,两种理论从不同侧面反映了大雷诺数湍流的流场结构.在本文第Ⅱ部分,我们将利用第Ⅰ部分中阐述的物理想法以适当方式建立两种理论之间的某种定量的联系.从而由拉格朗日观点的弥散运动的结果得出欧拉观点的结构函数、关联函数和能谱函数.所得结果不但适用于惯性子范围,而且适用于尺度更大(或波数更小)的全部范围.熟知的Kolmogoroff“2/3定律”和“(-5)/3定律”为本结果在惯性子范国的渐近解.因而本结果是Kolmogoroff“2/3定律”和“(-5)/3定律” 的推广.     

A fast level set model of multi‐atlas labels fusion for 3D MRI tissues segmentation with application of split Bregman method

In this paper, we propose a new fast level set model of multi‐atlas labels fusion for 3D magnetic resonance imaging (MRI) tissues segmentation. The proposed model is aimed at segmenting regions of interest in MR images, especially the tissues such as the amygdala, the caudate, the hippocampus, the pallidum, the putamen, and the thalamus. We first define a new energy functional by taking full advantage of an image data term, a length term, and a label fusion term. Different from using the region‐scalable fitting image data term and length term directly, we define a new image data term and a new length term, which is also incorporated with an edge detect function. By introducing a spatially weight function into the label fusion term, segmentation sensitivity to warped images can be largely improved. Furthermore, the special structure of the new energy functional ensures the application of the split Bregman method, which is a significant highlight and can improve segmentation efficiency of the proposed model. Because of these promotions, several good characters, such as accuracy, efficiency, and robustness have been exhibited in experimental results. Quantitative and qualitative comparisons with other methods have demonstrated the superior advantages of the proposed model.     

Lattice gauged principal chiral field at largeN

The lattice model of a principal chiral field interacting with gauge fields which have no kinetic term is considered. The complete set of equations for collective field variables is derived in the large N limit. The continuum limit and the phase structure of the model are discussed.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 25–31, July, 1995.     

Solutions of the Boltzmann equation with infinite energy: Existence, stability and long time behavior

This paper is devoted to studying a class of solutions to the nonlinear Boltzmann equation having infinite kinetic energy, these solutions have an upper Maxwellian bound with infinite kinetic energy. Firstly, the existence and stability of this kind of solutions are established near vacuum. Secondly, it is proved that this kind of solutions are stable for any initial data, as a consequence, the Boltzmann equation has at most one solution with infinite kinetic energy. Finally, the long time behavior of the solutions is also established.