Numerical analysis and computational testing of a high accuracy Leray‐deconvolution model of turbulence

We study a computationally attractive algorithm (based on an extrapolated Crank‐Nicolson method) for a recently proposed family of high accuracy turbulence models, the Leray‐deconvolution family. First we prove convergence of the algorithm to the solution of the Navier‐Stokes equations and delineate its (optimal) accuracy. Numerical experiments are presented which confirm the convergence theory. Our 3d experiments also give a careful comparison of various related approaches. They show the combination of the Leray‐deconvolution regularization with the extrapolated Crank‐Nicolson method can be more accurate at higher Reynolds number that the classical extrapolated trapezoidal method of Baker (Report, Harvard University, 1976). We also show the higher order Leray‐deconvolution models (e.g. N = 1,2,3) have greater accuracy than the N = 0 case of the Leray‐α model. Numerical experiments for the 2d step problem are also successfully investigated. Around the critical Reynolds number, the low order models inhibit vortex shedding behind the step. The higher order models, correctly, do not. To estimate the complexity of using Leray‐deconvolution models for turbulent flow simulations we estimate the models' microscale.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A family of new, high order NS-α models arising from helicity correction in Leray turbulence models

This report gives a reinterpretation of the NS-α model which leads to a family of high order NS-α-deconvolution models with NS-α as the zeroth order case. First, we show that the Navier-Stokes-α model arises by adding helicity correction to Leray-α model. Higher order Leray models have recently been proposed in [W. Layton, R. Lewandowski, A high accuracy Leray-deconvolution model of turbulence and its limiting behavior, Anal. Appl. 6 (1) (2008); W. Layton, C. Manica, M. Neda, L. Rebholz, Numerical analysis and computational testing of a high-accuracy Leray-deconvolution model of turbulence, Numer. Methods Partial Differential Equations, in press]: the so-called Leray-deconvolution models, that employ van Cittert approximate deconvolution to decrease consistency error. We use an analogous helicity correction idea to develop a family of higher order accurate NS-α type models, the NS-α-deconvolution models. We prove several mathematical and physical properties for this new family of models and discuss the design of efficient algorithms for them.     

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.     

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.     

Time decay rate for two 3D magnetohydrodynamics‐ α models

We consider the long time behavior of solutions to the magnetohydrodynamics‐ α model in three spatial dimensions. Time decay rate in L²‐norm of the solution is obtained. Similar results for a generalized Leray‐ α‐magnetohydrodynamics model are also established. As a by‐product, an optimal time decay rate for the Navier–Stokes‐ α model is achieved. Copyright © 2013 John Wiley & Sons, Ltd.     

Smoke flow phenomena and turbulence characteristics of tunnel fires

Gravity currents are similar in behavior with smoke flows. This work aims to provide evidence justifying the use of gravity current approach to model smoke flows downstream of the fire source. The turbulence solver available in almost all commercial CFD codes solves RANS for the flow field. To find out how well the nature of smoke flow be accurately modeled using RANS that is widely used for incompressible flows. The feasibility of using both Reynolds- and Favre-averaging schemes was numerically compared and examined in this paper. In this work, numerical simulations of a fire occurred in a 400-m longitudinally ventilated tunnel have been successfully performed using FDS version 4. Large eddy simulation is employed in this study. Although the ranges of fire size and ventilation velocity vary respectively from 0 MW to 100 MW and 0 m/s to 10 m/s, this paper focuses on the general flow and temperature fields and the turbulence characteristics. Furthermore, the turbulence kinetic energy levels of the flow in the tunnel at several locations were investigated. Since the flow field is generally induced by mechanical ventilation and combustion, the main contribution to the turbulence kinetic energy comes from its longitudinal, vertical, or their combination.     

A Navier–Stokes–Fourier system for incompressible fluids with temperature dependent material coefficients

We consider a complete thermodynamic model for unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional domain. The model comprises evolutionary equations for the velocity, pressure and temperature fields that satisfy the balance of linear momentum and the balance of energy on any (measurable) subset of the domain, and is completed by the incompressibility constraint. Finding a solution in such a framework is tantamount to looking for a weak solution to the relevant equations of continuum physics. If in addition the entropy inequality is required to hold on any subset of the domain, the solution that fulfills all these requirements is called the suitable weak solution. In our setting, both the viscosity and the coefficient of the thermal conductivity are functions of the temperature. We deal with Navier’s slip boundary conditions for the velocity that yield a globally integrable pressure, and we consider zero heat flux across the boundary. For such a problem, we establish the large-data and long-time existence of weak as well as suitable weak solutions, extending thus Leray [J. Leray, Sur le mouvement d’un liquide visquex emplissant l’espace, Acta Math. 63 (1934) 193–248] and Caffarelli, Kohn and Nirenberg [L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (6) (1982) 771–831] results, that deal with the problem in a purely mechanical context, to the problem formulated in a fully thermodynamic setting.     

Sur les équations α Navier–Stokes dans un ouvert bornéOn α Navier–Stokes equations on a bounded domain

We consider the Lagrangian averaged Navier–Stokes (LANS-α) equations in a bounded domain of $\text{R}{}^3$. We prove global existence and uniqueness of solutions under the hypothesis that the initial data belongs to H¹₀. To cite this article: A. Valentina Busuioc, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 823–826.     

Stability of the Crank–Nicolson–Adams–Bashforth scheme for the 2D Leray‐alpha model

We consider the stability of an efficient Crank–Nicolson–Adams–Bashforth method in time, finite element in space, discretization of the Leray‐α model. We prove finite‐time stability of the scheme in L², H¹, and H², as well as the long‐time L‐stability of the scheme under a Courant‐Freidrichs‐Lewy (CFL)‐type condition. Numerical experiments are given that are in agreement with the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1155–1183, 2016     

Non-linear k– turbulence model results for flow over a building at full-scale

Turbulence modelling is a crucial question in the application of CFD to flows over buildings. The impinging flow and anisotropic nature of the turbulence present severe challenges. This paper presents a comparison of CFD against full-scale results. It differs from previous work which has concentrated on the wind-tunnel scale. In order to better account for the production of turbulent kinetic energy and the anisotropic nature of the turbulence a non-linear k– model is implemented. The results are discussed for different turbulence models and for the comparison of computed results with the measurements from full-scale.     

On degrees of freedom of certain conservative turbulence models for the Navier-Stokes equations

We study the degrees of freedom of several conservative computational turbulence models that are derived via a non-dissipative regularizations of the Navier-Stokes equations. For the Navier-Stokes-α, the Leray-α and the Navier-Stokes-ω equations we prove that the longtime behavior of their respective solutions is completely determined by a finite set of grid values and by a finite set of Fourier modes. For each turbulence model the number of determining nodes and of determining modes is estimated in terms of flow parameters, such as viscosity, smoothing length, forcing and domain size. These estimates are global as they do not depend on an individual solution.     

Investigating Asymptotic Suction Boundary Layers using a One-Dimensional Stochastic Turbulence Model

The one-dimensional turbulence model (ODT) is applied to study turbulent asymptotic suction boundary layers for a Reynolds number of Re = u_∞/v₀ = 333, where u_∞ and v₀ are the free stream and suction velocity, respectively. In here we will demonstrate that a large eddy suppression mechanism may reduce the influence of ODT model parameters, such as the viscous cut-off parameter Z. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stabilità asintotica in media per moti fluidi viscosi in domini esterni

Summary In this paper we prove some asymptotic stability results for Navier-Stokes fluid filling an exterior domain. In particular, we show that if the Reynolds number associated to the basic flow is sufficiently small, then the energy of perturbations tends to zero as t,provided only that the energy is finite at t=0.This result also solves, as a special case, a problem pointed out by J. Leray in [25].

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Lavoro eseguito sotto gli auspici del G.N.F.M. (C.N.R.).     

Intermittency in flux driven kinetic simulations of trapped ion turbulence

Flux driven kinetic transport is analysed for deeply trapped ion turbulence with the code GYSELA. The main observation is the existence of a steady state situation with respect to the statistics, in particular the balance between the injected energy and the time averaged energy flowing out through the outer edge boundary layer. The temperature is characterised by a very bursty behaviour with a skewed PDF. Superimposed to these short time scale fluctuations, one finds a regime with a strong increase of the zonal flows and a quenching of the turbulent energy. During this phase of such a predator-prey cycle, the core temperature rapidly increases while the edge temperature gradually decreases. The end of this reduced transport regime is governed by the onset of turbulence that governs large relaxation events, and a strong modification of the zonal flow pattern.     

Single dopant diffusion in semiconductor technology

The paper deals with the analysis of pair diffusion models in semiconductor technology. The underlying model contains reaction‐drift‐diffusion equations for the mobile point defects and dopant‐defect pairs as well as reaction equations for immobile dopants which are coupled with a non‐linear Poisson equation for the chemical potential of the electrons. For homogeneous structures we present an existence and uniqueness result for strong solutions. Starting with energy estimates we derive further a priori estimates such that fixed point arguments due to Leray–Schauder guarantee the solvability of the model equations. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation of UV disinfection reactors: Evaluation of alternative turbulence models

Six turbulence models, including standard k–ε, k–ε RNG, k–ω (88), revised k–ω (98), Reynolds stress transport model (RSTM), and two-fluid model (TFM), were applied to the simulation of a closed conduit polychromatic UV reactor. Predicted flow field and turbulent kinetic energy were compared with the experimental data from a digital particle image velocimetry (DPIV). All of the predicted flow fields were combined with a multiple segment source summation (MSSS) fluence rate model and three different microbial response kinetic models to simulate the disinfection process at two UV lamp power conditions. Microbial transport was simulated using the Lagrangian particle tracking method. The results show that the fluence distributions and the effluent inactivation levels were sensitive to the turbulence model selection. The level of sensitivity was a function of the operating conditions and the UV response kinetics of the microorganisms. Simulations with operating conditions that produced higher log inactivation or utilized microorganisms with higher UV sensitivity showed greater sensitivity to the turbulence model selection. In addition, a broader fluence distribution was found with turbulence models that predicted a larger wake region behind the lamps.     

Numerical calculation of turbulent corner flows

Numerical predictions are presented of the hydrodynamic characteristics of developing and fully-developed turbulent flow in a square duct. The turbulent stresses in the plane of the cross-section, gradients of which cause the familiar secondary flows, are approximated by gradients in the axial mean velocity. Two distinct approximations are investigated, one of which specifies some of the model ‘constants’ as functions of the gradient of the length scale to account for wall effects. The stresses in the axial momentum equation are calculated from an eddy viscosity deduced from the K-W model of turbulence, K being the turbulence energy and W, a measure of the time-mean-square-vorticity fluctuations. The approximation incorporating wall effects generally performs better than the other when compared with fully-developed flow-data. This same approximation also compares favourably with data for developing flow and predictions based on K-? models in the literature.     

Weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies

We consider a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions and prove existence of weak solutions for it. In contrast to earlier contributions, we study a model with a singular nonlocal free energy, which controls the H^α/2-norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization.     

Existence of Weak Solutions Up to Collision for Viscous Fluid‐Solid Systems with Slip

We study in this paper the movement of a rigid solid inside an incompressible Navier‐Stokes flow within a bounded domain. We consider the case where slip is allowed at the fluid/solid interface through a Navier condition. Taking into account slip at the interface is very natural within this model, as classical no‐slip conditions lead to unrealistic collisional behavior between the solid and the domain boundary. We prove for this model existence of weak solutions of Leray type, up to collision, in three dimensions. The key point is that, due to the slip condition, the velocity field is discontinuous across the fluid/solid interface. This prevents obtaining global H¹ bounds on the velocity, which makes many aspects of the theory of weak solutions for Dirichlet conditions inappropriate. © 2014 Wiley Periodicals, Inc.     

On the introduction of the temperature standard in the undistinguishing parastatistics of objectively distinguishable objects

We consider various normalizations enabling us to change the scale of the graphs of isotherms and isochores. The relationship between parastatisticsal value of the maximal number of particles, corresponding to a given energy, and temperature allows us to pass in the domain of positive chemical potentials from the parastatisticsal number K (K = ∞ corresponds to Bose statistics and K = 0 to Fermi statistics) to the temperature, which changes the scaling of the pressure in this domain.