Multidimensional turbulence spectra – Statistical analysis of turbulent vortices

Strong nonlinear or very fast phenomena such as mixing, coalescence and breakup in chemical engineering processes, are not correctly described using average turbulence properties. Since these phenomena are modeled by the interaction of fluid particles with single or paired vortices, distribution of the properties of individual turbulent vortices should be studied and understood. In this paper, statistical analysis of turbulent vortices was performed using a novel vortex tracking algorithm. The vortices were identified using the normalized Q-criterion with extended volumes calculated using the Biot–Savart law in order to capture most of the coherent structure related to each vortex. This new and fast algorithm makes it possible to estimate the volume of all resolved vortices. Turbulence was modeled using large-eddy simulation with the dynamic Smagorinsky–Lilly subgrid scale model for different Reynolds numbers. Number density of turbulent vortices were quantified and compared with different models. It is concluded that the calculated number densities for vortices in the inertial subrange and also for the larger scales are in very good agreement with the models proposed by Batchelor and Martinez-Bazán. Moreover, the associated enstrophy within the same size of coherent structures is quantified and its distribution is compared to models for distribution of turbulent kinetic energy. The associated enstrophy within the same size of coherent structures has a wide distribution that is normal distributed in the logarithmic scale.     

槽道湍流中拟涡能输运对反向控制的瞬时响应

利用直接数值模拟研究了槽道湍流中脉动拟涡能输运对反向控制的瞬时响应． 发现流向和展向拟涡能的衰减首先由拉伸产生项的抑制引起,而法向拟涡能的减小是因为控制阻碍了平均剪切的倾斜．在控制的初始阶段,流向拟涡能的演化远远落后于其它两个分量的变化．法向涡量快速单调减小,并对其它两个分量的减弱起到了重要作用．     

A direct method to calculate the energy evolution of a turbulent flow

The classical configuration of the power spectrum of a homogeneous turbulent flow provides a functional relation between energy and enstrophy which may be used to find the evolution of energy in the intermediate stages of finite time and viscosity to which the usual asymptotic arguments do not apply. The method has a minimal computational cost compared with the direct numerical integration of the whole Navier–Stokes system. It is applied here to study the energy evolution both theoretically and numerically, obtaining the minima of energy compatible with the power spectra as well as the rate of decay of the energy towards them. The results may be useful to study the compatibility of the Kolmogorov and Kraichnan spectra with the observed energy evolution of a turbulent flow.     

Distributional Enstrophy Dissipation Via the Collapse of Three Point Vortices

Dissipation of enstrophy in 2D incompressible flows in the zero viscous limit is considered to play a significant role in the emergence of the inertial range corresponding to the forward enstrophy cascade in the energy spectrum of 2D turbulent flows. However, since smooth solutions of the 2D incompressible Euler equations conserve the enstrophy, we need to consider non-smooth inviscid and incompressible flows so that the enstrophy dissipates. Moreover, it is physically uncertain what kind of a flow evolution gives rise to such an anomalous enstrophy dissipation. In this paper, in order to acquire an insight about the singular phenomenon mathematically as well as physically, we consider a dispersive regularization of the 2D Euler equations, known as the Euler-\(\alpha \) equations, for the initial vorticity distributions whose support consists of three points, i.e., three \(\alpha \)-point vortices, and take the \(\alpha \rightarrow 0\) limit of its global solutions. We prove with mathematical rigor that, under a certain condition on their vortex strengths, the limit solution becomes a self-similar evolution collapsing to a point followed by the expansion from the collapse point to infinity for a wide range of initial configurations of point vortices. We also find that the enstrophy always dissipates in the sense of distributions at the collapse time. This indicates that the triple collapse is a mechanism for the anomalous enstrophy dissipation in non-smooth inviscid and incompressible flows. Furthermore, it is an interesting example elucidating the emergence of the irreversibility of time in a Hamiltonian dynamical system.     

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.     

Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation

This investigation is a part of a research program aiming to characterize the extreme behavior possible in hydrodynamic models by analyzing the maximum growth of certain fundamental quantities. We consider here the rate of growth of the classical and fractional enstrophy in the fractional Burgers equation in the subcritical and supercritical regimes. Since solutions to this equation exhibit, respectively, globally well-posed behavior and finite-time blowup in these two regimes, this makes it a useful model to study the maximum instantaneous growth of enstrophy possible in these two distinct situations. First, we obtain estimates on the rates of growth and then show that these estimates are sharp up to numerical prefactors. This is done by numerically solving suitably defined constrained maximization problems and then demonstrating that for different values of the fractional dissipation exponent the obtained maximizers saturate the upper bounds in the estimates as the enstrophy increases. We conclude that the power-law dependence of the enstrophy rate of growth on the fractional dissipation exponent has the same global form in the subcritical, critical and parts of the supercritical regime. This indicates that the maximum enstrophy rate of growth changes smoothly as global well-posedness is lost when the fractional dissipation exponent attains supercritical values. In addition, nontrivial behavior is revealed for the maximum rate of growth of the fractional enstrophy obtained for small values of the fractional dissipation exponents. We also characterize the structure of the maximizers in different cases.     

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.     

Maximal growth rate at the Rosensweig instability: theory,experiment, and numerics

We investigate the growth of a pattern of liquid crests emerging in a layer of magnetic liquid when subjected to a magnetic field oriented normally to the fluid surface. After a step like increase of the magnetic field, the temporal evolution of the pattern amplitude is measured by means of a Hall-sensor array. The extracted growth rate is compared with predictions from linear stability analysis by taking into account the nonlinear magnetization curve M (H). The remaining discrepancy can be resolved by numerical calculations via the finite element method. By starting with a finite surface perturbation it can reproduce the temporal evolution of the pattern amplitude and the growth rate. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Large-eddy-simulation of 3-dimensional Rayleigh-Taylor instability in incompressible fluids

The 3-dimensional incompressible Rayleigh-Taylor instability is numerically studied through the large-eddy-simulation ( LES) approach based on the passive scalar transport model. Both the instantaneous velocity and the passive scalar fields excited by sinusoidal perturbation and random perturbation are simulated. A full treatment of the whole evolution process of the instability is addressed. To verify the reliability of the LES code, the averaged turbulent energy as well as the flux of passive scalar are calculated at both the resolved scale and the subgrid scale. Our results show good agreement with the experimental and other numerical work. The LES method has proved to be an effective approach to the Rayleigh-Taylor instability.     

On the dynamics in a transitional boundary layer

IntroductionIll 1883 Professor Osborne Reynolds published in Philosopl1ical Transactions of the RoyalSociety the outcomes of his flow visua1ization at Manchester. These had shown that whetherthe flow in a pipe was direct to sinuous (or, as nowadays we would say, laminar to turbulent)depended on its Reynolds number. Transition from Iaminar to turbuIent flow becomes animportant probIem i1l fluid mechanics, which has attracted the interest of investigators fOrmore than l00 years. The partic…     

Local near-surface buckling of a system consisting of an elastic (viscoelastic) substrate,a viscoelastic (elastic) bond layer,and an elastic (viscoelastic) covering layer

Within the framework of a piecewise homogenous body model and with the use of a three-dimensional linearized theory of stability (TLTS), the local near-surface buckling of a material system consisting of a viscoelastic (elastic) half-plane, an elastic (viscoelastic) bond layer, and a viscoelastic (elastic) covering layer is investigated. A plane-strain state is considered, and it is assumed that the near-surface buckling instability is caused by the evolution of a local initial curving (imperfection) of the elastic layer with time or with an external compressive force at fixed instants of time. The equations of TLTS are obtained from the three-dimensional geometrically nonlinear equations of the theory of viscoelasticity by using the boundary-form perturbation technique. A method for solving the problems considered by employing the Laplace and Fourier transformations is developed. It is supposed that the aforementioned elastic layer has an insignificant initial local imperfection, and the stability is lost if this imperfection starts to grow infinitely. Numerical results on the critical compressive force and the critical time are presented. The influence of rheological parameters of the viscoelastic materials on the critical time is investigated. The viscoelasticity of the materials is described by the Rabotnov fractional-exponential operator. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 771–788, November–December, 2007.     

Cascade of energy in turbulent flows

A starting point for the conventional theory of turbulence [12–14] is the notion that, on average, kinetic energy is transferred from low wave number modes to high wave number modes [19]. Such a transfer of energy occurs in a spectral range beyond that of injection of energy, and it underlies the so-called cascade of energy, a fundamental mechanism used to explain the Kolmogorov spectrum in three-dimensional turbulent flows. The aim of this Note is to prove this transfer of energy to higher modes in a mathematically rigorous manner, by working directly with the Navier–Stokes equations and stationary statistical solutions obtained through time averages. To the best of our knowledge, this result has not been proved previously; however, some discussions and partly intuitive proofs appear in the literature. See, e.g., [1,2,10,11,16,17,21], and [22]. It is noteworthy that a mathematical framework can be devised where this result can be completely proved, despite the well-known limitations of the mathematical theory of the three-dimensional Navier–Stokes equations. A similar result concerning the transfer of energy is valid in space dimension two. Here, however, due to vorticity constraints not present in the three-dimensional case, such energy transfer is accompanied by a similar transfer of enstrophy to higher modes. Moreover, at low wave numbers, in the spectral region below that of injection of energy, an inverse (from high to low modes) transfer of energy (as well as enstrophy) takes place. These results are directly related to the mechanisms of direct enstrophy cascade and inverse energy cascade which occur, respectively, in a certain spectral range above and below that of injection of energy [1,15]. In a forthcoming article [9] we will discuss conditions for the actual existence of the inertial range in dimension three.     

Anomalous scaling in the model of turbulent advection of a vector field

We consider the model of turbulent advection of a passive vector field ϕ by a two-dimensional random velocity field uncorrelated in time and having Gaussian statistics with a powerlike correlator. The renormalization group and operator product expansion methods show that the asymptotic form of the structure functions of the ϕ field in the inertial range is determined by the fluctuations of the energy dissipation rate. The dependence of the asymptotic form on the external turbulence scale is essential and has a powerlike form (anomalous scaling). The corresponding exponents are determined by the spectrum of the anomalous dimension matrices of operator families consisting of gradients of ϕ. We find a basis constructed from powers of the dissipation and enstrophy operators in which these matrices have a triangular form in all orders of the perturbation theory. In the two-loop approximation, we evaluate the anomalous-scaling exponents for the structure functions of an arbitrary order. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 467–487, March, 2006.     

A theoretical model for Reynolds-stress and dissipation-rate budgets in near-wall region

A 3-D wave model for the turbulent coherent structures in near-wall region is proposed. The transport nature of the Reynolds stresses and dissipation rate of the turbulence kinetic energy are shown via computation based on the theoretical model. The mean velocity profile is also computed by using the same theoretical model. The theoretical results are in good agreement with those found from DNS, indicating that the theoretical model proposed can correctly describe the physical mechanism of turbulence in near wall region and it thus possibly opens a new way for turbulence modeling in this region.     

The Onset of Linear Instabilities in a Solid Combustion Model

This article concerns the onset of linear instability in a simple model of solid combustion in a semi-infinite two-dimensional strip of width l . The free boundary problem that describes the model involves initial and boundary conditions, including a nonlinear kinetic condition at the interface. The linear problem governing perturbations to a basic solution is solved by the method of images with the reaction front perturbation satisfying an integro-differential equation. This equation is then solved using Laplace transforms. Finally, we perform a stability analysis for the model by studying the solution of the reaction front perturbation. The inclusion of initial conditions enables us to show the development of linear instability from arbitrary initial small disturbances.     

基于有限分析法的平面铅垂壁羽流的相似解

应用计及浮力对湍动能及其耗散率的影响的k-epsilon湍流模式,结合有限分析法对密度差引起的平面铅垂紊动壁羽流进行了数值分析．在均匀环境条件下控制该类流动的连续性方程、流动方向的动量方程、浓度扩散方程、湍动能及其耗散率方程存在相似解．考虑到浮力通量守恒条件,应用有限分析法给出了铅直壁羽流的速度、相对密度差、湍动能及耗散率的分布,进而给出了各物理量最大值沿主流方向变化的关系式．湍流Schmidt数为1.0时的计算结果与实验资料吻合较好,表明应用有限分析法分析铅垂平面羽流是有效的,即在分析壁羽流时浮力对湍动能及其耗散率的影响应该予以考虑．     

Two-dimensional line thermal

The time evolution of a two-dimensional line thermal — a turbulent flow produced by an initial element with significant buoyancy released in a large water body, is numerically studied using the two-equation k-e model for turbulence closure.     

Construction of kinetic models for metabolic reaction networks: Lessons learned in analysing short-term stimulus response data

Construction of dynamic models of large-scale metabolic networks is one of the central issues in the engineering of living cells. However, construction of such models is often hampered by a number of challenges, for example, data availability, compartmentalization and parameter identification coupled with design of in vivo perturbations. As a solution to the latter, short-term perturbation experiments are proposed and are proven to be a useful experimental method to obtain insights into the in vivo kinetic properties of the metabolic pathways.

The aim of this work is to construct a kinetic model using the available experimental data obtained by short-term perturbation experiments, where the steady state of a glucose-limited anaerobic chemostat culture of Saccharomyces cerevisiae was perturbed. In constructing the model, we first determined the steady-state flux distribution using the data before the glucose pulse and the known stoichiometry. For the rate expressions, we used approximative linlog kinetics, which allows the enzyme–metabolite kinetic interactions to be represented by an elasticity matrix. We performed a priori model reduction based on timescale analysis and parameter identifiability analysis allowing the information content of the experimental data to be assessed. The final values of the elasticities are estimated by fitting the model to the available short-term kinetic response data.

The final model consists of 16 metabolites and 14 reactions. With 25 parameters, the model adequately describes the short-term response of the cells to the glucose perturbation, pointing to the fact that the assumed kinetic interactions in the model are sufficient to account for the observed response.     

An eigenvalue search method using the Orr–Sommerfeld equation for shear flow

A physically-based computational technique was investigated which is intended to estimate an initial guess for complex values of the wavenumber of a disturbance leading to the solution of the fourth-order Orr–Sommerfeld (O–S) equation. The complex wavenumbers, or eigenvalues, were associated with the stability characteristics of a semi-infinite shear flow represented by a hyperbolic-tangent function. This study was devoted to the examination of unstable flow assuming a spatially growing disturbance and is predicated on the fact that flow instability is correlated with elevated levels of perturbation kinetic energy per unit mass. A MATLAB computer program was developed such that the computational domain was selected to be in quadrant IV, where the real part of the wavenumber is positive and the imaginary part is negative to establish the conditions for unstable flow. For a given Reynolds number and disturbance wave speed, the perturbation kinetic energy per unit mass was computed at various node points in the selected subdomain of the complex plane. The initial guess for the complex wavenumber to start the solution process was assumed to be associated with the highest calculated perturbation kinetic energy per unit mass. Once the initial guess had been approximated, it was used to obtain the solution to the O–S equation by performing a Runge–Kutta integration scheme that computationally marched from the far field region in the shear layer down to the lower solid boundary. Results compared favorably with the stability characteristics obtained from an earlier study for semi-infinite Blasius flow over a flat boundary.