A mathematical and physical Study of multiscale deconvolution models of turbulence

This work presents a rigorous analysis of mathematical and physical properties for solutions of multiscale deconvolution turbulence models. We show that solutions of these models exactly conserve model quantities for the integral invariants of fundamental physical importance: kinetic energy, helicity, and (in two dimensions) enstrophy. The kinetic energy conservation is the key that allows us to next apply the phenomenology of homogeneous, isotropic turbulence to establish the existence of a model energy cascade and, in particular, that the cascade exhibits enhanced energy dissipation in a secondary accelerated cascade, which ends at the model's microscale (which we establish is larger than the Kolmogorov microscale). We also prove that the model dissipates energy at the same rate as true turbulent flow, ～ O(U³ ∕ L), independent of Reynolds number. Lastly, we prove the existence of global attractors for the model solutions; the proof of which also shows that solutions are actually one degree of regularity higher than previously known. Copyright © 2012 John Wiley & Sons, Ltd.     

Conservation laws of turbulence models

The conservation of mass, momentum, energy, helicity, and enstrophy in fluid flow are important because these quantities organize a flow, and characterize change in the flow's structure over time. In turbulent flow, conservation laws remain important in the inertial range of wave numbers, where viscous effects are negligible. It is in the inertial range where energy, helicity (3d), and enstrophy (2d) must be accurately cascaded for a turbulence model to be qualitatively correct. A first and necessary step for an accurate cascade is conservation; however, many turbulent flow simulations are based on turbulence models whose conservation properties are little explored and might be very different from those of the Navier-Stokes equations.We explore conservation laws and approximate conservation laws satisfied by LES turbulence models. For the Leray, Leray deconvolution, Bardina, and Nth order deconvolution models, we give exact or approximate laws for a model mass, momentum, energy, enstrophy and helicity. The possibility of cascades for model quantities is also discussed.     

An enhanced‐physics‐based scheme for the NS‐α turbulence model

We study a new enhanced‐physics‐based numerical scheme for the NS‐alpha turbulence model that conserves both energy and helicity. Although most turbulence models (in the continuous case) conserve only energy, NS‐alpha is one of only a very few that also conserve helicity. This is one reason why it is becoming accepted as the most physically accurate turbulence model. However, no numerical scheme for NS‐alpha, until now, conserved both energy and helicity, and thus the advantage gained in physical accuracy by modeling with NS‐alpha could be lost in a computation. This report presents a finite element numerical scheme, and gives a rigorous analysis of its conservation properties, stability, solution existence, and convergence. A key feature of the analysis is the identification of the discrete energy and energy dissipation norms, and proofs that these norms are equivalent (provided a careful choice of filtering radius) in the discrete space to the usual energy and energy dissipation norms. Numerical experiments are given to demonstrate the effectiveness of the scheme over usual (helicity‐ignoring) schemes. A generalization of this scheme to a family of high‐order NS‐alpha‐deconvolution models, which combine the attractive physical properties of NS‐alpha with the high accuracy gained by combining α‐filtering with van Cittert approximate deconvolution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On Taylor/Eddy solutions of approximate deconvolution models of turbulence

This article shows that so called general Green–Taylor solutions, also called Taylor solutions or eddy solutions, of the Navier–Stokes equations are also exact solutions to approximate deconvolution models of turbulence. Thus, these special structures in flows exist as exact features in the models studied and their persistence/transient behavior is exactly determined by their stability, not by the effects of modelling or truncation errors.     

Temporally regularized direct numerical simulation

Experience with fluid-flow simulation suggests that, in some instances, under-resolved direct numerical simulation (DNS), without a residual-stress model per se but with artificial damping of small scales to account for energy lost in the cascade from resolved to unresolved scales, may be as reliable as simulations based on more complex models of turbulence. One efficient and versatile manner to selectively damp under-resolved spatial scales is by a relaxation regularization, e.g. Stolz and Adams [S. Stolz, N.A. Adams, An approximate deconvolution procedure for large eddy simulation, Phys. Fluids II (1999) 1699-1701]. We consider the analogous approach based on time scales, time filtering and damping of under-resolved temporal features. The paper explores theoretical and practical aspects of temporally damped fluid-flow simulations. We prove existence of solutions to the resulting continuum model. We also establish the effect of the damping of under-resolved temporal features as the energy balance and dissipation and prove that the time fluctuations → 0 in a precise sense. The method is then demonstrated to obtain both steady-state and time-dependent coarse-grid solutions of the Navier-Stokes equations.     

Implicit subgrid‐scale modeling by adaptive local deconvolution

A class of implicit Subgrid‐Scale (SGS) models for Large‐Eddy Simulation (LES) is obtained from a new approach for the finite‐volume discretization of hyperbolic conservation laws. The extension of a standard deconvolution operator and the choice of a suitable numerical flux function result in a truncation error which can be forced to have physical significance. As determined by a modified‐differential‐equation analysis, the free parameters of this implicit SGS model can be adjusted to approximate a known explicit SGS model. Computational results for the viscous Burgers equation show that the model with parameters identified by evolutionary optimization gives significantly better predictions than other models. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear analysis of a parametrically excited beam with intermediate support by using Multi-dimensional incremental harmonic balance method

In this paper, a nonlinear Euler-Bernoulli beam under a concentrated harmonic excitation with intermediate nonlinear support is investigated. Continuous expression for the kinetic energy, potential energy and dissipation function are constructed. An energy method based on the Lagrange equation combined with the Galerkin truncation is used for discretizing the governing equation. The Multi-dimensional incremental harmonic balance method (MIHBM) is derived, and the comparisons between the numerical results and the approximate analytical solutions based on the MIHBM verify the excellent accuracy of the MIHBM. The steady state dynamic of the beam is investigated by MIHBM. In order to investigate the energy transmission and understand the vibration response of the Euler-Bernoulli beam, the effects of the key parameters on the dynamic behaviors are studied and discussed, individually. The results show that the amplitude-frequency curves exhibits softening nonlinear behavior in the super-harmonic resonance region, and near resonant region the hardening nonlinear behavior is observed depending on the different parameters. Nonlinear dynamic analysis, such as bifurcation, 3-D frequency spectrum, waveform, frequency spectrum, phase diagram and Poincaré map, are also presented in order to study the influences of the key parameters on the vibration behaviors for the beam in a more accurate manner. In addition, the path to chaotic motion is observed to be through a sequence of the periodic motion and quasi-periodic motion.     

Grid filter models for large-eddy simulation

A new interpretation of approximate deconvolution models (ADM) when used with implicit filtering as a way to approximate the projective grid filter is given. Consequently, a new category of subgrid models, the grid filter models (GFM) is defined. ADM appear as a particular case of GFM since only approximate deconvolution is achieved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reduced-dissipation remapping of velocity in staggered arbitrary Lagrangian-Eulerian methods

Remapping is an essential part of most Arbitrary Lagrangian-Eulerian (ALE) methods. In this paper, we focus on the part of the remapping algorithm that performs the interpolation of the fluid velocity field from the Lagrangian to the rezoned computational mesh in the context of a staggered discretization. Standard remapping algorithms generate a discrepancy between the remapped kinetic energy, and the kinetic energy that is obtained from the remapped nodal velocities which conserves momentum. In most ALE codes, this discrepancy is redistributed to the internal energy of adjacent computational cells which allows for the conservation of total energy. This approach can introduce oscillations in the internal energy field, which may not be acceptable. We analyze the approach introduced in Bailey (1984) [11] which is not supposed to introduce dissipation. On a simple example, we demonstrate a situation in which this approach fails. A modification of this approach is described, which eliminates (when it is possible) or reduces the energy discrepancy.     

On Energy Cascades in the Forced 3D Navier–Stokes Equations

We show—in the framework of physical scales and \((K_1,K_2)\)-averages—that Kolmogorov’s dissipation law combined with the smallness condition on a Taylor length scale is sufficient to guarantee energy cascades in the forced Navier–Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws—in terms of Grashof number—for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.     

Cosmological inflation models in the kinetic approximation

We consider the construction of cosmological inflation models with an approximate linear dependence of the kinetic energy of the scalar field on the state parameter. We compare the obtained solutions with known cosmological models and calculate the main parameters of cosmological perturbations.     

Computation of energy dissipation in visco-elastic materials at finite deformation

In this contribution, the derivation of the energy dissipation rate in generalized visco-elastic material models with internal stress-type variables and linear evolution equations is outlined. The approximated dissipation rate is computed from a positive quadratic form of the nonlinear non-equilibrium stresses and the inverse of the consistent material tangent tensor. The presented method is used to compute the energy dissipation of visco-elastic rubber material in a large scale application of a steady state rolling tire structure. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kinetic decomposition for kinetic models of BGK type

In this paper, we consider kinetic models of BGK type which describe the scalar conservation law at the microscopic scale. We use new technique developed in Comm. Partial Differential Equations 27 (2002) 1229 in order to get the convergence. First, we obtain the approximate transport equation for the given kinetic models of BGK type. Then using the averaging lemma, we obtain the convergence. This paper shows how to relate the given kinetic model with the averaging lemma to get the convergence.     

Approximate Deconvolution Models for Magnetohydrodynamics

We consider the family of approximate deconvolution models (ADM) for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove the existence and uniqueness of solutions to the ADM-MHD equations, their weak converge to the solution of the MHD equations as the averaging radii tend to zero, and derive a bound on the modeling error. We demonstrate that the energy and helicity of the models are conserved, and the models preserve the Alfvén waves. We provide the results of the computational tests, that verify the accuracy and physical fidelity of the models.     

On the existence of an inertial manifold for a deconvolution model of the 2D mean Boussinesq equations

We show the existence of an inertial manifold (ie, a globally invariant, exponentially attracting, finite‐dimensional manifold) for the approximate deconvolution model of the 2D mean Boussinesq equations. This model is obtained by means of the Van Cittern approximate deconvolution operators, which is applied to the 2D filtered Boussinesq equations.     

Modeling and numerical simulation of multiscale behavior in polycrystals via extended crystal plasticity

The purpose of this work is to exploit the algorithmic formulation of models for multiscale inelastic materials whose behavior is influenced by the evolution of inelastic microstructure and the corresponding material or internal lengthscales. The models for extended crystal plasticity are based on the formulation of rate potentials whose form is determined by (i) energetic processes via the free energy, (ii) kinetic processes via the dissipation potential, and (iii) the form of the evolution relations for the internal-variable-like quantities upon which the free energy and dissipation potential depend. Examples for these latter quantities are the inelastic local deformation or dislocation densities as GNDs. Different algorithmic implementations are discussed, namely the algorithmic variational approach and the dual mixed approach. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An explicit finite‐difference method for the approximate solutions of a generic class of anharmonic dissipative nonlinear media

We develop an explicit finite‐difference method to approximate solutions of modified, Fermi–Pasta–Ulam media, which consider the presence of parameters, such as external damping, relativistic mass, a coefficient for the nonlinear term, and a coefficient of coupling in the case of discrete systems. We propose discrete expressions to approximate consistently the total energy of the system and the average energy flux, and prove that the discrete rate of change of energy is a consistent estimate of its continuous counterpart. The method is thoroughly tested in the energy domain, and our results show that the method gives an approximately constant energy in the case of conservative systems, which fluctuates within a narrow margin of error that may be attributed to truncation errors. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Approximating the larger eddies in fluid motion V: Kinetic energy balance of scale similarity models

We first review a classical scale-similarity model used to simulate the motion of large eddies in a turbulent flow. The kinetic energy balance of this model is very unclear in theory. Experiments with it often have reported that an additional Smagorinski type subgridscale term is needed. This term is not benign; it can alter significantly the predicted long term dynamics of the large eddies. However, we also show that the principal of scale-similarity (introduced in 1980 by Bardina, Ferziger and Reynolds) can also give rise to other scale similarity models which have the correct kinetic energy balance.